The Pokemon types I use the most? Hmmm...that's a tough question. Since I have never actually battled competitively, since 1. I've never actually been able too, and 2. I simply couldn't be bothered with the breeding and EV training and such like that it seems you have to do if you want to be a good competitive battler, I don't have any favourite Competitive battle types.
However, since I do play the games, I do have favourites for the In-Game Pokemon, mostly the Fire, Water, Psychic and Dragon Types. After Gen II, Steel and Dark were added.
To wit: all my main Pokemon in ANY game has always had this formula of:
Fire (mostly the fire starters)
Water (mostly the water starters or legendary, as in the case of Suicune)
Psychic (Espeon, Alakazam, Gardevoir are the main ones I use)
and Dragon (Dragonair, Altaria, Flygon etc...).
With the other two, Dark (either Umbreon or Honchkrow)
and Steel (Steelix, Lucario and Bastidon)
Of course, I also use other types, and have used pretty much all of them except for Poison, Bug or Grass, since I really am not fond of them. Fighting I also had an aversion to, until Blaziken, Infernape and Lucario came along, then I liked them.