Who's responsibility is it to spray for bugs?

What kind of bugs and when was/is the infestation? Were they in the house before you moved in or did you acquire them since you've taken occupancy?

(In California, and hopefully where you are), It is your landlord's responsibility to provide safe and habitable housing, including freedom from insects and vermin, basically rats and roaches. However, check your lease as it might have a clause that you are to conduct the pest control or are to maintain a clean rental. The regulations require that the landlord compensate you or conduct the pest control if your place is messed up because of poor maintenance that is the landlord's fault.

I don't know what the rules say if you are the one causing the insect problem. Enforcement very likely depends on the intensity, cause, fault, and pest. The health inspector isn't likely to check your flea problem if you own dogs, but might be interested if the next door neighbor is letting the roaches get out of hand

Check your state's health and safety code or consult with your local health department. In California two rules that apply are California Health & Safety Code Section 17920.3 and California Civil Code 1942.1.

If it's your landlord's fault, or if it is your neighbor's fault who also rents in the same building, and if it is a serious problem, definitely ask him to get it fixed or compensate you to get it fixed. If he refuses, well... If your landlord caused the problem, he/she has likely been asked to fix the problem before you lived there.

I checked my lease. I agreed not to violate any law... not to commit, suffer or permit any waste, or nuisance in, on or about the said premises... or annoy, molest any other tenant. (I wonder if my wife could be included;) Specifically, I agreed to prevent any flea or other infestation, caused by my cat.