There are 3 primary uses for 3 1/2" 12 gauge, waterfowl (mainly geese), turkeys, & coyotes. Any game that you would consider using a 10 gauge for the 12 gauge 3 /1/2" is a decent choice.
I have owned 3 different 3 1/2" 12 gauge shotguns. A Mossberg 835 with a 28" barrel, a Mossberg 835 with a 24" barrel in woodland camo, and a Benelli Super Black Eagle LH. I actually had the SBE setup for Turkeys with a 1-4 Leopold scope and pistol grip stock. It was also a good setup for coyote, but I never got the opportunity to use it on one.
I also had a Browning BPS 10 gauge.
For Turkeys a liked the 3 1/2" 12 over the 10 gauge because of the lighter gun weight. I have killed several Turkeys with the 3 1/2" and a couple with the 10 gauge. I love to run and gun for gobblers and the weight difference of the shotguns makes the 3 1/2" 12 a ideal choice for the big bangers. In a goose blind, 10 gauge please: as the weight of the shotgun eats up the recoil.
Mentioning recoil the 3 1/2" 835 Mossberg 24" is the most recoil that I have ever endured!!!
If you are a woods coyote hunter and use hounds, the additional pellet count in a 3 1/2" #4 buckshot load will roll a yote out to 70-75 yards.
My current turkey shotgun is a Western Field 410 bolt action shotgun

. The fun of calling in a Tom to close range is more fun than rolling them at 60+ yards with the Benelli Ultimate Turkey blaster!
If you are on a limited budget and can only justify one shotgun. The 835 Mossberg with the 28" barrel is truly a great do it all shotgun. Start your day with 3 1/2" loads in the goose blind. After filling your limit of geese. Head to the drainage ditches and jump shoot a few ducks with 3" steel. On the way back to the truck load up with 2 3/4" light loads and catch a few migrating timber doodles.
There is no shotgun that offers more versatility than a 3 1/2" gauge, and the 835 is probably the best value in the bunch.