Are bean sprouts kosher for Passover?

It depends. If you are of Sephardic or Mizrahi extraction then yes, they are. If, however, you are Ashkenazi, then traditionally no, they're not. This is because they fall into the category of kitniyot, small things - peas and beans from which it is possible to make flour that could become confused with flour made from the Five Grains. If a Jewish person eats any food in which any of these grains has come into contact with water for more than eighteen minutes during manufacture, the food becomes chametz and he or she risks karet - being spiritually cut off from G-d, one of the harshest punishments in the Jewish tradition and consciousness.

However, the ban on kitniyot is a rabbinic tradition aimed at helping people avoid accidentally condemning themselves to karet - it does not have a religious reason and is not included in the Torah. For that reason, many Ashkenazi Jews - especially those who stick to a vegetarian or vegan diet, as I do, for whom legumes are an important source of protein and a major part of what they eat - do now eat kitniyot during Passover, though we take very great care to ensure we are not eating chametz.