I'm most familiar with Storm/Roto Grip and Brunswick, so my suggestions will follow suit.

 

Strong Arc: Probably want a symmetrical core here, although you can get an assymetric core for an early hook/set reaction to mimic an "arc" reaction. I have a Virtual Gravity sanded to 360 grit which stands up at the arrows and sets in the midlane. But this is sometimes a hard reaction for your ball driller to mimic. So here are some symmetrical core suggestions. These would be good for longer sport/PBA patterns where you need to get the ball into an early roll. (Scorpion)

 

Pin down: Storm Marvel, Brunswick Wild Card

 

Strong Skid/Flip: Personally, I like symmetrical cores because of their continuation, but if you want more of a flip reaction, then go with an assymetric core. The key is to get the ball to flip in the right spot downlane, to maximize carry.  Possibly options for this would be as follows. These would be good for medium to long sport/PBA patterns which require you to get the ball through the heads due to the lower volume of oil. (Shark)

 

 Pin up: Storm Virtual Gravity, Brunswick C-System Alpha Max (good deal with the $50 rebate offer)

 

Medium Arc: Benchmark ball, many possibilities here. I like to use symmetrical cores for benchmark type reactions. These would be good for reading a pattern, and also medium length patterns which require you to play straighter up the track area of the lane. (Viper, Chameleon, Scorpion) 

 

Pin down: Storm Hy-Road, Storm Reign of Fire (take the surface down to 4000), Brunswick Diamondback, Roto Grip Riot, Brunswick Anaconda

 

Medium Skid/Flip: Probably a pearl or hybrid assymetrical core, but again symmetrical cores are more versatile and continuous. Again, many options. Good for moving left and opening up patterns once they begin to break down.

 

Pin up: Storm Anarchy, Storm Prodigy, Roto Grip Mutant Cell Pearl, Brunswick Wicked Seige (again good deal with $50 rebate offer)

 

Weak Arc: Definitely want a weak symmetrical core for this reaction, and possibly put an extra hole on your PAP to remove differential (which makes the ball less reactive downlane). Also, it may be necessary to add light surface to even out the reaction. These options would be good on shorter sport/PBA patterns. (Cheetah, Viper)

 

Pin down: Storm Natural, Storm Tropical Heat Hybrid, Brunswick Avalanche Solid, Columbia Freeze

 

Hope this helps.