Valentine's Day Mew
A big part of fandom seems to be deciding who is in love with or who should be in love with some other character. This activity, called shipping (as in, relationships), can sometimes celebrate obvious pairings that might even be officially endorsed by the source material. Sometimes shipping explores might-have-been or could-be relationships, even if they are not considered canon. And at the extreme, shipping can include characters that have never even met, sometimes going so far as to crossover with characters in other story universes, fan characters, or real-world people!

The core of my fandom activities, personally, involve exploring the canon (even if it involves silly loophole exploration and cognitive dissonance from attempting reconcile conflicting versions of a story), and I am not that interested in other people's love lives, so I've never been much of a shipper. I understand shipping is all in good fun and I can respect that, even if it is not really my cup of tea. It's just odd to me, as a non-shipper, that not even Mew and Mewtwo are immune to this sort of matchmaking. Forgive me if I am skeptical, but here are the shippings, their slight proofs, and some discussion on what I think about them.

Pairing Mew and Mewtwo Together

Ew, stop flying so close to me!  I don't want your cooties!

Some people think Mew and Mewtwo are in love. This shipping is usually called mewshipping, but I have also seen it referred to as catshipping.

One reason to believe this shipping is that, at the end of Mewtwo Strikes Back, Mew and Mewtwo don't seem too upset about flying next to each other. Fans popularly believed that Mew led Mewtwo and the other clones to a safe, secret place to live in harmony (even before Mewtwo Returns). In addition, the two psychic cats usually appear together during the opening theme songs, often at least tolerating the other's presence. Finally, fans often seem to make the argument of, if not for Mew, who else would be close enough to Mewtwo's "species" to be a viable match for him? (well, hold on for the other shippings...)

Hey, personal space, whoa!

It seems the usual response to Mewshippers, from other non-believing fans, is the question of if such a pairing is in some way incestuous. If you follow the games' canon, you could interpret the story to mean that Mew is indeed Mewtwo's literal mother. In other canons, Mewtwo was cloned from Mew's DNA, which would mean Mew and Mewtwo are roughly genetically identical. A counter-argument would be that the scientists sliced and diced Mewtwo's genetic code so much that he no longer so closely related to Mew anymore. Or, Mewtwo was cloned from a fossil remnant of an ancient Mew that was not the same individual as the Mew(s) living today, so their genetic codes are different enough for them to no longer be "family".

Mew, you're so hot, you are literally on fire!  Burn, baby, burn!

I have also seen fans debating over whether Mew and Mewtwo's genders or physiology would affect their ability to breed, and therefore the likelihood of this pairing. I want to be clear that, er, the possibility for successful breeding is not strictly necessary for a romantic relationship. However, even without genetic reasons, Mewtwo probably still has plenty of revulsion for the idea of a romantic relationship with Mew, and vice versa.

One of the main themes of Mewtwo Strikes Back and the prequel Drama is Mewtwo dealing with the inferiority complex from being "merely" a clone. He was haunted by dreams of Mew, who taunted or dismissed him, and his primary goal, even as he stepped out of the flames of the destroyed laboratory where he was "born" minutes earlier, was the annihilation of the source of his DNA, thereby proving his superiority and justifying his existence. Mew, likewise (at least in the Japanese version), believes that clones are innately inferior to the originals, and I speculate the only reason Mew traveled to New Island in the first place with the goal of eliminating such aberrations. While both Mew and Mewtwo are presumably over their respective complexes now, I don't think their turnarounds will be so quick that they'll be falling head over heels for each other any time soon. Actually, they probably both need extensive therapy.

Comic by Psylady

I also happen to think Mew and Mewtwo's personalities and life experiences are different enough that it would take a great deal of time for Mewtwo to even have a friendship with Mew. Mewtwo likes to be still and think. Mew is probably very wise, but still likes bouncing on bubbles and riding windmills and spinning around. Mewtwo's past is dark, and especially marked with manipulation by those that he trusted and by the death of his best friend. Mew, as far as we can tell, has always been free, and, as an immortal creature, possibly has no real understanding of death. I think the last thing Mewtwo needs is a Manic Pixie Dream Girl (a hyperactive, wacky chick that teaches the brooding emo loner how to enjoy life - a common and ridiculous romantic comedy trope). I do believe Mew and Mewtwo could get along to some degree and learn a lot from each other, but I personally don't see it becoming anything more than friendship. But hey, I don't want to burst your psychic pink bubble. If Mew or Mewtwo are both immortal, perhaps anything could happen in a couple of centuries!

Pairing Mewtwo with Ai / Amber

Come on, they're babies.
Other people like to pair Mewtwo with his childhood friend and fellow clone Ai (dub name Amber), who appears in stories of Mewtwo's Origin. This shipping is called Cloneshipping, Copishipping, or Replicashipping. Think this pairing needs more names? You can also call it what must be one of the most coveted of shipping names: Loveshipping (since the name "Ai" also means "love").

This shipping is very weird to me, as it seems to take Mewtwo and Ai's sweet friendship and twist it. First of all, they are basically babies. During the CD Drama, Ai is approaching her fourth birthday in her life as a clone. Ai watched as Mewtwo was created, so Mewtwo must be even younger than that! These are toddlers; they don't have an adult-level interest in romance yet! They themselves view their friendship in a sister-brother or mother-son type way.

Oh, it's so cute! That's Myuutsuu...Can I be its mama? Can I be its older sister? I wouldn't mind living if I have a child like that. Hello Myuutsuu.
Ai The Birth of Mewtwo CD Drama Episode 2: The Birth of Myuutsuu

Maybe childhood friends end up staying best friends or even get married. Or maybe they end up growing up into different people and live different lives. You can't really judge the relationships of kids that haven't even been potty-trained yet, expecting them to find the person they will want to spend the rest of their lives with at age three. I am not saying Ai and Mewtwo aren't adorable, and that their friendship wasn't beautiful or special. Yes, they do love each other, but as best friends. There is no need to insist that this dear friendship was in any way romantic. Friendships are precious in and of themselves, and not every relationship needs to be shoehorned into a pairing.

The empty tank

Could it have been a possibility for the future? Well, maybe. But unfortunately that future can never happen. The individual Ai that Mewtwo knew (whether she was Aitwo or Aithree or Aifour) did not survive the cloning process and died. She is gone forever.

To get around the whole death issue, I have seen some fanworks that retcon or ignore that part of the story entirely, or instead simply focus on the time while Ai was still alive; other fanfics address the death by presenting the idea that Ai was cloned again. However, cloning does not resurrect an individual; it creates a new individual that happens to have the same set of genes. The Pokémon universe adds in a fanciful idea that clones have vague memories of the lives of those they were cloned from, which would not be true in real life, but even allowing for these hazy snippets of "past lives", another clone of Ai might not even remember Mewtwo, and she could easily have a completely different personality. Another clone would be more like a twin sister of Ai, not the same individual. If you think cloning could save Ai and allow for her to have a future, you are making the same mistake as her father, Dr. Fuji.

Make them? As many as you want? That's not right. There's only one Ai. [...] There was one Ai who talked to me.
Myuutsuu The Birth of Mewtwo CD Drama Episode 4: The World's Strongest Pokémon
Fading away

The only concession I can make is the idea that Ai lives on inside Mewtwo's heart. Mewtwo was made to forget the time he had spent with Ai, and one of the saddest things is that there is no solid evidence that he remembers the girl he once was friends with at all. However, I believe that his friendship with Ai did leave a permanent impression upon him, even if he cannot consciously recall it. That powerful but intangible love is the only part of this shipping that I personally can agree with.

Pairing Mewtwo with Nurse Joy

Standing next to each other = Love

Still other people think Mewtwo likes Nurse Joy, and sometimes refer to this belief as Newislandshipping or Crossshipping. True, Mewtwo whisked Nurse Joy away to his own private island, but it was really nothing romantic. Mewtwo did so because he needed her knowledge of Pokémon, and she served as a good human face to use when contacting unsuspecting Pokémon trainers. I personally believe there was nothing more to it than that, and I certainly didn't see any flicker of undying love in his eye when he released her from the trance. Don't forget Mewtwo was on a "I hate humans!" kick at the time. I really don't feel there will be any developing emotions in the future, at least not positive ones. Joy's memory of the events was erased, but if she were to remember it, she might need to attend the therapy sessions as well...

Pairing Mewtwo with Giovanni

Giovanni's ego-stroking session with Mewtwo

Pairing Mewtwo and Giovanni is called Abuseshipping or sometimes Devastationshipping. While I must say the dynamic between them is strong and interesting, if you are referring to a romantic relationship, I can't really agree with that. Giovanni is obsessed with Mewtwo, but Giovanni does not love Mewtwo: he loves Mewtwo's power and how Mewtwo's power will benefit him. Giovanni has manipulated, imprisoned, and tortured Mewtwo. When Mewtwo first encountered Giovanni, he trusted him and did his bidding, thinking he had found a "partner" who would not treat him as if he were an inferior copy. Mewtwo's idea of what was normal was warped by Giovanni so that the Pokémon saw everything as a battle and prized strength as the most important quality. Once Mewtwo realized his trust had been betrayed, and that Giovanni merely saw him as a tool or a slave, he not only hated Giovanni, but all humanity as well as their Pokémon (assuming they were all similarly brainwashed and enslaved by their trainers). Mewtwo bore the scars of Giovanni's abuse throughout Mewtwo Strikes Back, and took a long time to realize that everything he had learned (consciously or sub-consciously) from Giovanni was wrong. When Giovanni found Mewtwo again in Mewtwo Returns, his only goal was to reclaim his investment in his secret weapon, no matter how much suffering Mewtwo endured in the process.

If anything, Mewtwo and Giovanni's relationship, as the name says, is an abusive relationship. Being abused is not a sign of love, and if you are in an abusive relationship, you must seek outside help and remove yourself from the situation as soon as possible (blasting through the sky not necessarily required).

Pairing Mewtwo with Sabrina

Sabrina being all psychic

Some people think Mewtwo and Sabrina, the Saffron City Gym Leader and keeper of the Marsh Badge, are destined for love. This actually even has names: Psyshipping or Psychicshipping. For clarification, I am pretty sure this shipping is only considering the old-school anime version of Sabrina, and not the Team Rocket member from the manga or the pop-star idol she has become in the games. But, thinking of the cursed psychic with the creepy ability to turn people into dolls, I can see how they are similar, as they are both serious, skilled, and seem to be loners. Perhaps, with their shared psychic abilities, they would be able to understand each other, and thereby eventually fall in love, though, er, the "loner" thing might make that difficult. Plus, Mewtwo and Sabrina have never actually met. I could see them perhaps being friends in the future, if they ever do cross paths.

Pairing Mew with Ash

Spiny Spin

The idea that Mew and Ash are in love is called Dreamshipping, and it seems to be based mostly on the idea that it would be cute, heh.

However, some after-the-fact rationalizations exist: Mew (who had previously avoided notice at New Island) revealed itself in order to save Ash's life, and, in the end, Ash was the only one who saw Mew flying away, even though Brock and Misty were standing right beside him. I personally think these are moreso due to Ash's "Main Character" status than any romantic feelings Mew may or may not have for the twerp, but there you go.

If you specifically pair Ash with the Mirage Mew from the special Mirage Pokémon episode, some fans refer to that as Safeshipping, if you'd rather be more specific.