87' Honda prelude si will start and then cut off. All the lights on the dash board comes on, then cut off.

with a probe light get into back of dash board and see where power is or isnt

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Your starter motor is either not getting power or in need of replacement. Otherwise, there is nothing wrong with your car.

Also check your battery connections for no corrosion there as well as by the starter motor. Low voltage circuits in a car are extremely vulnerable to poor contact issues.

sounds like the speed sensor

if you can charge the battery and drive it for a while then sounds like the altenator has failed and is not re-charging the battery, good way to test is to charge the battery, start the car and turn the light on, if everything is dimm or the motor dies there is little to no charge voltage, and will require altenator work

Try changing the Transmission oil - can solve a host of problems, including rough changes.

Just fixed this myself. I found a used USDM meter/gauge out of a 1995 SI on EBAY for $200. However, I have a VTEC engine that redlines at 7500 RPM and the SI's redlines at 6500 RPM. Easy to fix by swapping the face plates inside of the meter/gauge. If your issue is the bulbs they cost a lot... around $250 each. But if your lights went out simultaneously then the bulbs are not the issue... it's the power supply where the bulbs plug into and that's located behind the drivers dash speaker (very easy to remove with only two screws holding it in). This part is more expensive at around $350, but cheaper than buying two bulbs. Scroll through every page of honda prelude listings on EBAY and you may just get lucky like I did. Hope this helps.

head light switch is bad

my 93 prelude was doing the exact same thing and for me it was my alternator was working properly.
go to a parts store and have them test it.

Its probably the timing belt... mine did the same thing