Although I've played a couple DS games on my 3DS, I usually end up playing most of them on a DS system. The battery life is longer and the system is less heavy than the 3DS along with the fact that there are minimal benefits to playing a game on the 3DS are all reasons why backwards compatibility hasn't been as big a feature as I thought it might be.
Generally, I'm sticking with the DS Lite when playing DS games unless the game has a larger or heavier touchscreen focus like Feel the Magic, Pac-Pix or Kirby Canvas Curse. In cases of games like those, then I'll play them on my DS Phat to protect the touchscreen for my DS Lite a bit more. But, like UncleBob mentions, due to the better screens of the DS Lite, I'm always a little saddened to do this since I know the colors and look will be a bit more muted on the Phat. I remember after playing through Canvas Curse, I popped it in to the DS Lite just compare how it looked and it was so much brighter and colorful. Even the sound on the speakers was a bit better.
I do like my Phat but after that comparison it's hard to go back to it. I should double check but I think I found the contours of the DS Lite better for playing Mario Kart DS as well for all the L and R button pressing one needs to do in that game. I suppose I should look into screen protectors for the DS Lite touchscreen and then I'd really be able to retire the system.
The only other benefit I consider it having over the Lite is that GBA games and the Rumble Pak fit snuggly into the system while they stick out a bit on the Lite. Not that the Rumble Pak does much. A Muffled Alarm Indicating Rumble Is Occuring Pak is a better description for it but I've still tried to use it for games that allow the option. As for GBA games, having a bit of a bigger screen to play them is nice and it was on the Phat that I played Metroid Fusion and Zero Mission for the first time giving me better memories of the system but nowadays if I am playing a GBA game then the GBA Micro is my first choice. Haven't really done a screen comparison between the Micro, SP, Phat or Lite to see how games look like on all of them but my guess is that the Lite would probably be the best option or second best option on the off-chance the Micro is the best just like DS games. Maybe someone here can share their thoughts on that bit of info.