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Does a 96 have a vehicle speed sensor commonly referred to as a VSS? I'll bet it does unless it has a cable for the speedo in which case check the speeedo cable.
Due to me being a dumbass, I had the same issue a few years ago with an 05 Softail Standard. I dropped the bike while getting off of it and somehow ended up cutting a wire in the wiring harness. I was lucky since it was under the seat behind the battery. I'm not saying that is what is wrong with your bike, but if it isn't anything else, check the wiring.
I had the same problem earlier this year on my 99. It turned out to be the voltage regulator. Changed it out and no problems since. You can test your regulator easily with a trouble light on the plug. Good luck
I had an 02 Fatboy that did about the same thing, only speedo was intermitent/dead and turn signals dead. Traced it down to connectors under tank console, used steel wool/emory cloth and coated with silicone grease and it fixed my problem.
You might just want to replace the Sensor. Mine went bad after just 20k miles on my 97. Also when the Speedo isn't working your turn signals won't self-cancel. Once the Speedo is working again, your signal will start self-canceling again. It's a cheap, quick & easy fix as long as you don't bring it to a dealer.
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