I'm currently stocking up on Heal Balls, just because of their color I love Net Balls too. If only I could buy Premium Balls instead of getting them as a freebie... They look so much better than the regular Poké Balls!
I did use the Apricorn Balls in Gen II (and a bit in SS), except for the Friend and Moon Balls, though I still don't really get the point of the Moon Balls. Do we REALLY need a specific PokeBall that works best on Moon Stone-evolving Pokes? I don't know; it never made much sense to me. Also, I don't really find much use in Dive Balls after Gen III (with the Net Balls available) and Heal Balls seem a bit useless to me since I don't get them until AFTER I'll already have a full team on-hand with me.
In Gen III, I didn't know where the Premier Balls came from and I had gotten 3 of them that I saved (again, I didn't know you just get them when you buy 10+ Poke Balls) on my first run-through of Sapphire. Since I thought they were special, I decided to try to use them to catch the 3 Legendary golems. I can't remember if I succeeded or not though....
I do use special balls every once in a while now, more so in Gen IV and the post game of Gen V. I never liked that by the time I got around to getting the special ones, I pretty much had a set team already in Poke/Ultra Balls (never use Great Balls that much), so I never have much variety in animation. I do mostly use whatever I have now to catch any Pokes to fill my 'dex, especially the Nest Balls. I have been using Repeat Balls quite often recently. I do love the Dusk Balls though and use those a lot more often than any other special ones. If I have a few of any kind of special balls to spare that I just want to get rid of, I'll just throw out whatever I have.
Legendaries I try to get in Ultra Balls, though I will use Fast/Quick Balls at the beginning (espically roaming legendaries like Entei) or Timer Balls later on during battles when I am SLOWLY whittling down health. Sometimes, I will catch Pokemon with theme balls, such as Audino with a Heal Ball. Most of the time, I just use Ultra Balls (or Poke Balls if I 'm feeling daring), though I tend to stay stocked up on several different varieties just in case I need them or have the urge to use them for some reason.
I am usually strict on using only Pokeballs for my captures, however there were times that I liked to use a special ball like a Heal Ball, Dusk Ball on a Pokemon that I favoured. For example I caught a Mienfoo in Victory Road with a Heal Ball because it felt like a special capture, as if I knew I would be using this Pokemon a lot in the future.
And ever since my Mienshao has been the best lead to a party that I have ever used. I also think that it makes the Pokemon seem more special and unique in a way.
Only if they're available in the Pokemarts, I can't be bothered collecting apricorns. If the situation to use a particular ball occurs, then I will give them a try, but I don't really trust them to work better than Ultra Balls. Dusk Balls in Gen IV are about the only ones that had a noticeable difference, but that's because they were slightly broken.
I use almost all kinds of pokéballs, usually depending on the colors, looks and types of pokémon. for example:
I use heal balls on cute pokémon like chansey and jigglypuff; dusk balls look perfect on dark- and ghost-type pokémon such as gengar and weavile; dive balls for water-type pokémon; I use the net ball specifically to catch surskit.
I never use great balls because they look so ugly.