Pokemon x Pokemon ships?
For me, it's more of a per-character case than anything else with little regard to the specie... well, except in "unique character species" (COUGH!LegendariesCOUGH!) cases.
Still... here are a few of those I use and how I usually portray them.
Mewtwo (male) x Mew (female) (Mewshipping): My single most overused one. Yes, Mewtwo is Mew's clone... so what? Considering the DNA ratio (I mean... just look at Mewtwo!), the two are technically unrelated despite Mewtwo's origins. Regardless of this factor, the two are opposites but in a pleasant way. Their awkward yet so strange connection make them work just as well as lovers as enemies... and sometimes, as both. A definitive favorite of mine.
Mewtwo (male) x Celebi (female): Almost if not always an one-way relationship. Although "destined" to be together, Mewtwo has no intention of following Fate and the beautiful, manipulative. and twisted Celebi will do everything, even the lowest deeds, to get what she desire more than anything else... Mewtwo's love.
Lucario (male) x Mew (female): Anime and Mystery Dungeon say that Lucario is a Legendary. D/P say that Lucario is the Rattata of Shinou. I opt for the former case, thank you. Anyway, that aside, this is probably my second favorite with Legendaries. Lucario, the servant of Mew, is totally dedicated to his master in ways that go beyond duty... and Mew might be more receptive to him than he might want to imagine. Yet, because of the relationship's nature, the relationship could turn out bittersweet or even a distraction from their respective duties.
Jirachi (male) x Mew (female): Another "destined" couple... and this one has no love going for it actually. Despite the strong attraction that bind the two, Mew would rather have a male that care for her and Jirachi is not interested in relationships. Yet, this has potential to be surprisingly cute and contradicting. Faking out love or faking out indifference? Who knows?
Lugia (male) x Ho-oh (female): This one is a rather original one. The two creators of the world and first-born Legendaries (Aruseus? Hahaha...), they have been together since, like, forever. When the Legendaries also gave themselves genders and feelings, they also were among the first to receive those and to hook-up with each others. This couple is far from being your usual shippy one ; this is a couple that is litterally older than dirt and that fell out of passion a few billions years ago. As such, the two act like an old couple, arguing and never getting along too well. Yet, despite obviously being unable to endure each others, they stick together. Habits die hard, it seems...
Latios (male) x Latias (female): Yes, they're siblings. Unlike for Mewtwo x Mew, no fancy DNA excuse here. Yes, it's creepy. It's unbelievably cute and disgustingly sweet despite everything.
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Well, for Legendary shippings, that's pretty much it. As for regular species... it's more or less a case per case pattern more than species but well, a few stand out...
Charmander (male) x Totodile (female): For some reason, this one turned out incredibly cute and sweet despite the fact that the Charmander line is strongly overused and that the Totodile line isn't the most "matchable" specie ever.
Combusken (male) x Kirlia (female): One of my favorite Pokemon x Pokemon ships that do not involve Legendaries if not my favorite.
Blaziken (male) x Gardevoir (female): I will be frank ; while I like Torchic and Combusken, I believe that Blaziken is probably one of the ugliest Pokemon in existence. I also see Gardevoir as usually little more than a flying dress with a cheap human-like puppet head on it. Still, despite all that, that particuliar ship just stand out. Dunno why, really.
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Well, as a general rule, those are the Pokemon x Pokemon ships I use the most. Dunno if those outside of the first one have names...
Have a nice day.
