You have a starter/ignition switch mounted in the steering column and you have a starter solenoid switch that is mounted to the starter.
check your ignition switch.
Bad neutral or clutch safety switch? Bad starter? bad starter solenoid?
If ignition problems still persist after replacing the neutral safety switch on a 1995 Chevy S10 check the starter. A broken starter will result in a car which will not be able to turn over the engine.
The starter relay feeds power to the starter when the switch is turned to start. The relay will be mounted either on the fender or on the firewall.
Starter solenoid switch is stuck closed.
probably the ignition switch if you are refering to the starter solenoid.
you have to remove the starter to replace the relay and the relay cost almost as much as the whole starter so you are better off just replacing the whole starter.
the blinker relay is behind the glove box
Seized engine? Loose or corroded battery cables? Weak or dead battery? Bad starter solenoid? Bad starter? Bad clutch or neutral safety switch? Bad ignition switch?
the starter relay.
Contact you Chevy dealer there was a TSB (technical service bulletain) for the s10 there is a problem with the ignition switch. I drive a 2003 S10 and have had the same problem. I am not sure if this repair is free but it could be the answer to your problem. Obviously you should go this route if the battery, starter and all other solutions have been checked and tried.