Before checking the flasher, check the fuse and bulbs first. Most likely the fuse is out or one of the bulbs is out. It could be corrosion on a bulb socket too. After that, look at the flasher relay. If it ends up being a bulb, see sources and related links below.
hazards and turn signals have separate flashers. replace flasher for turn signals
It has a flasher relay for the signals only.
Need to know what year S-10 you have.
Defective flasher module - common problem
Aren't they different flasher relays
The flasher for the turn signals is located on what Chevrolet calls the convenience center. It is located behind the dash to the left of the steering column. It is anything but convenient to get to.
Defective flasher relay.
it is behind the glove box.
There is a flasher for the turn signals installed under the dash on the driver side. It will be just to the right of the steering column.
The flasher is mounted just to the left of the steering column. The turn signal and emergency flashers are built together on this model.
There are 2 flasher relays. One for the turn signals and one for the hazard lights. The Turn Signal flasher is located on the drivers side, passenger side of steering column, under dash mounted near cigarette lighter. Optionally it can be mounted under the dash on the drivers side of the steering column. The Hazard flasher relay is mounted in the exact same locations.
The flasher relays are in the back of the glove box the fuses are on the left side of the dash and can only be accessed if the driver's door is open