What we usually refer to chilli in Singapore is really but one particular type of chilli pepper that is available here in the US. I'm still not sure what the proper name for it is, but some research points to cayenne pepper being the right thing.

Mexican cooking uses various forms of chillies, and these are the ones you'll generally find in supermarkets (i.e. Giant Eagle?). However, most of them (jalapeno, habanero) don't have the right kind of spicyness so you can't substitute them for good old chilli.

Fresh Red/Green Chilli

You can get red chilli from the Giant Eagle? Market District in Shadyside, at the peppers section. Green chilli seems to be more widely available at other Geagles, but I've not tried them before so I don't know if they taste right.

Often you can substitute fresh chilli in a recipe using dried chilli by just soaking the dried chilli in hot water for several minutes till it gets soft.

Dried Chilli

Several places stock dried chilli; Lotus Supermarket? and other places in the Strip probably have some. Kholil's? has dried chillies too.

Chilli Padi

Chilli Padi is the small little super-hot chilli that you most often see with the soy sauce when you got to eat in Chinese restaurants (in Singapore at least). I think its English name is "tabasco pepper", but again I'm not fully certain. This can be found at Lotus?, and probably several Giant Eagle?'s as well.