In terms of design, I love the Moon, Luxury, Premier, and Cherish Balls. They're all really pretty, and I love the animation used for the Luxury/Premier/Cherish Balls when a Pokemon is sent out of them.
In terms of usage, Timer, Quick, Dusk, and Ultra Balls. Timer and Dusk Balls have helped me capture many a legendary; Quick Balls have helped me catch some shinies and some legendaries, as well as Pokemon I need for breeding (don't wanna spend too much time or balls on a Pokemon I'll never use on my team, after all); and Ultra Balls help when the other three can't be used effectively.
If I could create a specialized Poke Ball, it'd be one that works better if the Pokemon is inflicted with non-volatile and volatile status conditions. I'd call it the Status Ball, and it'd have a mix of a hazy/fiery/cloudy design. As for the way it works:
Say, for example, a Pidgey is burned. Its chance of being caught is multiplied by 1.5 because of it. Because it is inflicted with one status condition, the Status Ball's rate is x2 instead of x1; thus, the capture rate is at x3. However, you decide to use Confuse Ray on the Pidgey and confuse it. The Status Ball's rate is now x3. The capture rate this time is x4.5. The more status inflictions it has, the higher the Status Ball's capture rate goes; however, it'd increasing in capture rate at four status conditions (where it'd be at x5).
I guess it's a bit complex, but it'd still be interesting nonetheless~