What type of rat are they having issues with and are they living in the shed or just entering it looking for food? I'd first find the entrance areas and address them then worry about catching any remaining rats. Norway rats can have tunnel systems at the base of the shed while roof rats are generally going to be above ground.
I've never used a gopher trap for rats, but perhaps someone on here can say if they've had success with them in the borrows or not.
In general, any trap that works for a squirrel will in theory work for a rat as well. For cage style traps you can use are the Sherman (metal style) straps, chipmunk traps, or small squirrel traps. You could use a coon trap if you wanted but the key really isn't the trap as much as the wire size. The best is solid, then 1/2 by 1/2, with 1 by 1/2 the largest I'd recommend.
With body gripping traps you can use #40, #50, or #55, RGB 3" round. or even some of the Koro traps. Based on your laws, snares are even a possiblity with a thin wire that closes all the way. Of course, glue boards are an option most think of but I've never had success with them and don't put a lot of faith in them.
With foodholds, I've heard of guys successfully body catching them in #2s (similar to mink I'd think), but again I've never done it and have never seen it.
Another option is to move the snap traps and place them into bait boxes. Since you said multiple traps have been tripped, it is possible that the traps aren't positioned correctly for how the rats approach them and the bait boxes would help with the issue.
