For me, Paladins have always been my favorite. There is something cool and heroic about riding into battle on a horse with a lance. They are also a very versatile class that can be used for offense and defense. It's a class full of cool and interesting character like Marcus, Geoffrey, Percival or Sain.
The aesthetics of the Bride class, from their dress to their acrobatic attack animations. I also like their weapon selection a lot; Bows and Staves are a combination I really like, and it gets extra points for being an unmounted, unarmored lance-wielder.
Official Bride and Wife of Noire (of the Fire Emblem Awakening message board)
Assassin is a classic for me. I always love Assassins/Speedy classes that don't suck (*cough*Swordmasters from FE7 onwards*cough*).
But recently, Bow Knight became my favorite class. Take a look at the concept art some time, you'll see it is the only mounted female class in the game with a functional breastplate and pants. I think the aesthetic takes after those old Japanese horsemen. Really cool. It has my favorite weapon combination (Sword and Bow) and it is mounted. What's there not to love?
FE series makes me like the Wyvern Rider class too. It seems Wyvern Riders have a tendency to be really good units. Altenna, Miledy, Haar, Heath, and Jill off the top of my head all kick lots of ass. My FE13 waifu is a Wyvern Rider too (Cherche). Cherche would be great if only she had f***ing Galeforce rage rage rage.
".....It's fine, Mr. Wright... Even a bluff would suit me just fine... As long as you can... ..clear up this doubt inside me!" -Apollo Justice
Grandmaster... because of those Badass Longcoats. I also like how they use both swords AND tomes; the whole Badass Bookworm feel just really gets me. Also, their Skills are just really badass, too. Don't forget their Pair Up bonuses... they can make just about any partner into an instant badass.
Grandmasters, in general, are just badass.
Great Lords take a close second because their outfits are as f*cking majestic as Xerneas itself.