depends where you are.

In many areas, roaches are the biggie, esp. in crowded areas. In really old buildings, there might be mice (I had both in my NYC prewar rental apt, but only roaches, and that very rare, in my NYC co-op)

Otherwise, really no different kinds of pest than you'd have to watch out for in a house--ants, etc.

NYC has bed bugs lately, but you pick them up in weird places, usually hotels.

I was worried about that before we moved into this apt., because I had been living in a house. So before we moved in I used the bug bomb; roach motel and put out rat poison. This place had new carpeting; new kitchen flooring and was freshly painted and professionally cleaned. I didn't ever get any bugs in the traps or laying down with their little feet up in the air. I take my garbage out every nite; don't leave dirty dishes in the sink, or cookies out. Yet other people in this building complain about bugs. But I went into somebody's kitchen and they had a paper bag for the trash (no can) and it was overflowing with shrimp shells at 10am.
They complained the bugs were terrible. I guess bugs know where to go for dinner.
Talley is right about the bed bugs, the Wall Street Journal said even the fanciest hotels have that problem.

Is there a law where you live about places needing to be funigated every 6 months or something like that? I know there is where I live. Here if they don't fumigate at the regular interval the town sends someone out and does it and the building owner has to pay the bill. Also we can go and look up when the last fumigation was and if there were any problems with certain pests etc. Just a thought to look into.