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I have a 1996 Honda Accord that will not start.  The battery is OK.  No dim lights.  The fuses are OK.  The starter does not even make a sound. What do I check next?  

I am having the same problem with my 91 Accord. Starts with a jump,runs fine but will not start on its own. Unhooked the battery cable,waited 5 mins and all lights, cd player have full power. Turn the key everything dies. I wonder if its the main relay? HEEEEEEEEEEEEELP  

My 1996 Honda Accord sporadically did not start with no sounds (no reving or clicking). I replaced the battery when it was acting up and a jump started it one time. Then it still didn't start on occassion so I replaced the starter. STILL it did not start whenever it felt like it (no time of day or temperature association).

So I just replaced the ignition switch just the other day. If this doesn't fix it, I don't know what will! I recommend starting by replacing the ignition switch because, although it was a little more abrasive, it was the least expensive and seemed to do the trick.  

I HAVE A SAME PROBLEM... I DON'T KNOW WHAT TO DO!!  

I had a similar problem for my 98 accord.  Honda mechanic replaced the ignition switch and the problem is gone.  Costed $48 for ignition switch and $150 for labor.  

Ok I have a 1996 Accord it wouldn't hit a lick no click no nothing. Swapped the battery nothing. I also have no lights or power of any kind except the horn and the break lights work. Can't roll up or down windows no air the stereo has no power nothing. Someone said starter. Well my son was broke down out of town and someone crossed out the starter bypassing the solenoid and got it started some how. I replaced the starter because you can't get the solenoid separate. This done nothing. I was baffled. I was thinking Ignition because turning the Key over once didn't get power moving. Before I broke off into this I started checking relay's. using a paper clip I removed the battery relay located under the hood on the passenger side in the corner fuse box next to the fire wall. It didn't look bad but I used the paper clip to bridge the gap like a "n" ding ding ding it got power. Ok at this point I've put the $109 starter on and have never cranked it. I was going to take it off and get my money back on it but I thought I better take the old one and have it tested since it's free and it's off. I'd hate to take the new one off and put the old one on and find out it was bad also. The old one failed. I had both issues. The relay for some reason allowed the starter to burn up before blowing. The relay was an 80 amp. The one I got at auto zone didn't have the legs bent like the other. with needle nose we bent the legs the right way and trimmed off some of the excess copper so it fit and it worked. The relay was $3.30. Try this and I bet it works for you. Maybe you should start with the battery relay first. You may get out of a starter. I hired this done by a shade tree mech. that knows how to work on cars for $30.00. It was 98 degrees today and I'd paid twice to not have to jack with this. The starter was hard to get to but most of the work was done from the top. Took about an hour total. Good luck hope this helps someone out.
Jeff  

I know this is an old post, but I just recently went through this same problem on my 1996 Accord V6 as described.  Mechanics said the battery then the starter.  My symptoms led me to believe that it was neither, but they told me that's all it could be.

When I drove the car to work, it started up fine.  Then when I returned, it would not even try to start.  I got a vibrating electrical sound from around the gauges and ignition switch and once in a while I could get a solenoid click.  I noticed that the clock would  reset back to 12:00 PM and the lights would go off for a short time and then come back on bright (at least in the daylight)

AAA was able to get it started, but he tried to sell me a battery after saying he thinks the battery was ok.  After I drove home, I tried to restart it, same thing, clicking sounds but it would not start.  I cleaned the connections to the battery and made sure the grounds and wires were tight.  I replaced the battery.  Same problem.

After reading online I checked the fuses and other modules, they checked out fine.  I found an online mechanic and he said it could only be the starter or a blown engine.  I am able to spin the engine by hand, so I knew it was not blown.  He said that it had to be the starter and nothing else, so much for relying on a paid online mechanic!    I changed out the starter, same problem, won't start.

After a little language exchange with the car I started all over checking everything.  I still believed this was a power problem, so inspected the electrical connections and ground wires again, this time closely.  I made sure they were tight again, but as I approached the negative terminal I  noticed some green corrosion under the nut, so this time I removed the nut and the wire connection from the terminal.  I found the problem, the connection was fully corroded under the nut and the as it touched the terminal. Everything else looked clean.  After cleaning the wires initially and putting some anti-corrosive grease on them, I thought they were good to go.  I removed the wire from the terminal and cleaned it well, then I put it back on the negative terminal and connected the battery.  All is well, no more problems.  I will replace the ground wire and I will keep the old battery and return the starter and next time I won't assume that I cleaned the terminals good enough!

I HOPE THIS WILL HELP SOMEONE ELSE IN THE FUTURE.