Johto: I've always liked the Love Balls, Moon Balls, and Friend balls. They had such a cute design.
Hoenn: Timer Balls, and Luxury Balls. Timer balls are amazing when going against legendaries and it might take a while to catch them. Luxury balls have a cool little design, and give a happiness boost, so that's a plus.
Sinnoh: Cherish Balls and Dusk Balls. Basically for design reasons.
Not sure. I never made much usage of special balls since there weren't too many legends running laps all over the whole region left and right in my Gold ROM, and since I played Ruby before any of the other Gen 3 games, I was too much of a newb to find much more to use than the three standard Pokeballs and Premierballs. FRLG was even more limited since the only special balls you acquire are things that you need to bring specific Pokemon for, but in later generations, I used a bit more than that.
In terms of design, I guess I like the Ultraball, Repeat ball, and Duskball. Since mass-cloning Masterballs never get old, I'd have liked it if they gave a better "entrance ceremony" when something pops out of it in Gen 4, as it is the ball that has no peer (well, save balls in the Palpark ). The purple flash just doesn't do it for me, as it was among the highlights of the generation before.
In terms of function, since you were always limited to a single ball per day from Kurt, I never made much use of the special balls (and can't with my Gold version's battery goin' out), so that's one generation out.
Third gen, Nestballs, Great balls, and Ultraballs are all good,
In gen 4, I used Greatballs, Ultraballs, and Duskballs most often, and sometimes Repeat balls.
Normally I use a combination of Dusk Balls and (in situations where those are no good) Ultra Balls. Most other types I use pretty infrequently, though I do use Net and Dive Balls reasonably often since there are so many water types.
However, once a game has a completed Pokedex, I switch from Ultra & Dusk to Repeat & Dusk, since by that point Repeat Balls have a 3x catch rate in all possible circumstances instead of Ultra's 2x.