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An s-10 transmission with no reverse most likely has an internal problem. The shifter fork or a gear issue can cause no transmission reverse.
look into wiring plug on transmission (under area of transmission). I know this for a fact on manual transmission. auto transmission- i would check in service repair manual.
There are several things that can cause your manual transmission not to shift out of first gear or reverse. The most common cause is a broken shifting fork.
Reverse
Reverse
Reverse
Reverse
Reverse
Transmission problems.
Reverse
Manual, top of transmission. Auto, inside the transmission.
Probably the reverse switch is bad or out of alignment. You'll need a manual to find the reverse switch. It'll be somewhere on the shifter, on the transmission, or possibly somewhere in between.