You have only to play at Little Wars three or four times to realise just what a blundering thing Great War must be.Although H.G. Wells notes that other wargames pre-dated his Little Wars, I think that most historical wargamers are willing to credit him with being one of the early founders of their hobby. Wells made no pretense that he was doing anything other than playing with toy soldiers, and to this day, there are wargamers who continue the tradition of unabashedly playing with toy figures.
Great War is at present, I am convinced, not only the most expensive game in the universe, but it is a game out of all proportion. Not only are the masses of men and material and suffering and inconvenience too monstrously big for reason, but — the available heads we have for it, are too small. That, I think, is the most pacific realisation conceivable, and Little War brings you to it as nothing else but Great War can do.
– Little Wars
Lego Minifigures are probably the most popular due to their versatility and customizability. Rules systems such as BrikWars, YALUGS, and others have been specifically created for Lego wargaming.
Playmobil figures are also popular, but probably come in at a distant second place. These figures are the closest in appearance to classical toy soldiers, and are suitable for both historical and fantasy gaming. The only Playmobil specific system I've seen is Three Inch Glory.
Petite figures differ from Lego and Playmobil figures in that by their nature, the majority of them are personalities. This makes them more suited for skirmish level gaming, unless you ignore some of their unique characteristics and assign them generic roles in a large army.
In any event, two of the systems that I think would go well with petite figures would be SBH from Ganesha Games for skirmish level rules, and HOTT from Wargames Research Group for larger scale battles. Neither system has anime specific rules, but they are flexible enough to simulate the genre with a bit of work.