The two most commonly-available corys are the Corydoras aeneus and Corydoras paleatus, both also have an albino form; these can manage in medium hard to hard water with a pH in the high 7's.
A few others, such as Corydoras sterbai, Corydoras panda and Corydoras trilineatus are often available, and should fare OK, although C. panda can be quite sensitive.
Most other species if available will be wild caught, and need soft to medium hard water, pH preferably below 7 but in the low 7's some may manage. Many of these are in our profiles, under catfish, if you'd like to check them out. There may be some that would be OK with harder water; this will be indicated under the preferred water parameters for each species.