My Miata's a-c works and it's cold, but after about 30 minutes of driving, the a-c starts to squeak and make noises. What's going on here? Thanks.

 

Mine is doing the exact same thing. I've been told it needs a replacement fan belt, but I'm not sure how much something like that would cost.

 

I would check the belt as gnome suggests but also check the fan operation and a/c pressures. The compressor is quite a load to run but if a fan quits or the pressures get too high it becomes even harder to turn. Depending on what year it is they also had a problem with the a/c compressor bracket to block bolts falling out and causing a loose compressor which can potentially damage the crankshaft keyway and snout.