My favorite apps are the following:
Digital Clock. I always used it to check and keep track of time when I wasn't using another app.
Pedometer. Very useful for keeping track of how many steps you had left to walk while recharging the VS. Seeker.
Friendship Checker. Great way to check the Friendship levels of Pokemon that evolve through Friendship.
Day Care Checker. I used it while breeding to check when Eggs were created.
Marking Map. Very good for chasing down roaming legends, and marking places on the map if there were some areas where I had unfinished business to take care of later on.
Some others I kind of like, but never used that much, were these:
Dowsing Machine. Used it a few times to find hidden items, but it wasn't that often. I prefer the Dowsing Machines in Gen 5 and HG/SS.
Berry Searcher. Useful for planting berries, but I pretty much never planted any berries in these games, so it wasn't that useful.
Counter. I have heard that many players use this for counting EVs, but I just used my traditional pen and paper for that instead.
Analogue Clock. I prefer the Digital Clock since it shows time more accurately.
Coin Flip. I used it sometimes if I couldn't make a choice on my own, and for comfirming my seed while RNGing.
Colour Changer. Used it to change the Poketch color to blue, but that's about it.