Sounds like a dirty and/or loose connection to me. First one I would check is battery to chassis ground cable connection. After checking all other connections try this procedure.
I wrote this procedure with tractor style mowers in mind but this works on zero turn mowers as well. For a zero turn mower, the steering levers must be out, parking brake engaged and the PTO switch in the off position. Some mowers have an operator presence switch in the seat and you may need an assistant to sit in the seat while performing the following tests.
Electrical problems can be very easy or very difficult, depending on four things:
1. How well you understand basic electricity.
2. What tools you have and know how to use.
3. How well you follow directions. Ahem.
4. You don't overlook or assume anything and test/verify everything.
First you need to check every wire and connection, making sure they are not dirty, corroded, broken or loose. *Failure to do this can lead to inaccurate reading or tests down the road.
Remember, I cannot see what you are doing. You are the eyes, ears and fingers in solving this problem. You must be as accurate as you can when you report back. The two basic tools I will ask you to use are a test light and a multimeter. If you have an assistant when going through these tests it would be very helpful. These steps work the best when done in order, so please don't jump around. Now let's solve this problem.
First, make sure the parking brake is engaged and the PTO switch is in the off position. Physically remove the negative battery cable first and then the positive battery cable and clean both the battery terminals and cable connectors with a wire brush. Reinstall cables after cleaning starting with the positive first and then the negative. Next, turn the key to the run position, check all fuses with a grounded test light should light on both sides of fuse. Check battery voltage - above 12.5 volts should be good. Believe it or not, this first step will take the most amount of time, usually around five minutes and the rest of the steps can be accomplished in under a minute.
Second, check for power from the battery to one of the large terminals on the solenoid. One of the wires is connected directly to the battery and has power all the time so one of the large terminals should light a test light or show battery voltage on a meter at all times.
Third, check for power at the small terminal of the solenoid while depressing the clutch/brake pedal and holding the key in the start position (you may need an assistant to sit in the seat to override the safety switch). If your solenoid is a four wire solenoid, check both small wire terminals as one is ground and the other is power from the ignition switch. If your solenoid is a three wire solenoid, make sure the solenoid body is not corroded where it bolts to the chassis of the mower as this is your ground path back to the battery. If in doubt, remove the solenoid and clean the mounting area down to bare metal. If there is no power to the small terminal then your problem is most likely a safety switch, ignition switch or in the wiring. You should have battery voltage on this small solenoid wire when the key switch is held in the start position.
Fourth, check for power on the other large terminal of the solenoid while holding the key in the start position (you may need an assistant to sit in the seat to override the safety switch). You should have battery voltage on this terminal when the key is held in the start position.
Fifth, check for power at the starter while holding the key in the start position (assistant again). You should have battery voltage during this test.
Sixth, check your ground circuit back to the battery. Just like the battery, make sure connections are clean and tight. Very Important!
This procedure is a simple starting point and there is more to it so after you have gone through each of the above steps, let me know what happened when you performed each test. At that point I will have great info to tell you how to proceed. Remember you are the eyes, ears, and fingers, so please be as accurate as possible. Some lawnmowers use a relay in the starter control circuit so keep that in mind.
Be as specific as possible with voltage readings as this will help diagnose your problem quicker. If you do not know how to perform the above checks, just ask and I will try to guide you through it. I prefer to use voltage drop tests but some people get too confused over this subject so in an effort to keep it simple, for now, just follow the procedure and report back with your findings.
