Went for a short ride and when I got back my turn signals (all of them) weren't working. All the other electronics on the bike work. From what I've heard this eliminates the fuse since the GS500 only has one (??).

Where should I start troubleshooting?

Thanks! 

Amy

Check for voltage on the orange and the light blue wires at the turn signal relay. (With the key in the on position, turn signal switch off)
You should have 0 volts between orange and lt.blue, and 12-14 volts from each of those to the battery negative terminal.
If you have power on the orange, but not lt.blue, replace the relay.

Short out the blue and orange on the connecter going into the relay with a paperclip or something similar, then flick the switch to the the left and right, if the bulbs on each side light up the relay is toast.

Where is the relay located?

Under the right side plastic, just below your thigh as you're sitting on the bike.  Small black cube.

Thanks for the advice! I found the relay, shorted out the contacts and the turn signals stay lit like you mentioned. The component is a:

Guilera Electronic flasher
12v 21+21+3,4W
012340S
38610-01D10

Can I just use an automotive flasher or do I need this specific model?

Thanks

Cant say for certain, I have always swapped like for for like. I guess if you fit a non OEM part designed to handle 45W at 12v it should work  Try one that you know is servicable and see if it works, it wont do any harm to your bike.

The bottom figure in the list is the Suzuki part number