This may get long, it may be confusing, you may get a headache. If you aren't ready for this, why are you here? Go grab a cup of coffee and lets get with it.

First consider the obvious. Battery, how old is it, is it charged. Make sure your electrical power source is what it should be before doing anything else. Due to vibration and irregular usage, rider batterys are doing pretty well to last much more then a couple of years. Don't expect to much from them.

Make sure you as the operater of the equipment, are doing what you should. There are safety switch's all over riders that prevent starting or cranking to help avoid injuries. Make sure you have the clutch/brake pedal fully depressed, make sure the shifter is in neutral, make sure you are sitting in the seat, make sure the blade control is turned off. Make sure there is oil in the engine. Yeah, that low oil shut down on many engines, on some units it's a no crank sensor.
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OK, everything good to this point? It better be or you shouldn't be here yet. Don't be trying to fry the egg before the chicken has laid it!
There are seven basic power cable connections you need to check next, followed by 3 additional sub connections. First and last before any of the following. First remove the ground terminal at the battery. Last, reconnect the ground terminal at the battery. This will prevent accidental shorting while working on the remainder of the cables. It is also the first one to check and clean.

Now the rest. Positive battery terminal(2). Negative battery cable to ground/frame(3). Positive cable in to starter solenoid(4). Positive cable out of starter solenoid(5). Positive cable to starter motor(6). Engine ground to frame(7). Engine ground to frame is normally through direct bolt down, make sure mounting bolts are tight. Some, will have rubber mounting bushings for anti-vibe purposes, these should have a ground strap from block to frame.

Remove each cable from it's terminal, clean and reconnect making sure it's tight.
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Sub connections.
Typically from the battery in, at the starter solenoid, will be a fused wire running up to the key switch. This may lead from the large battery in or from a smaller terminal near the battery in of the starter solenoid. This wire may or may not pass through an amp gauge on the way to the starter key switch on the dash. Make sure the connection at the solenoid is clean and tight. And hey, check that in line fuse while you're at it.(1)
The return wire from the key switch to the starter solenoid. This "WILL" go to a small terminal near the large out cable of the stater solenoid. Clean and tighten.(2)
The wire harness plug in at the key switch. Unplug, spray carb or electrical cleaner in and out of the plug and all over the terminals of the key switch. Dust and dirt build up can cause poor contacts.(3)

Now you have cleaned and tightened all of the major connections of power. Last, re-connect the battery ground cable.
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Are you still with me? Ready for that second cup of coffee?
Since you're still reading this I guess that means you still can't get the starter to crank. So get out your trusty volt meter or continuity light and lets take some readings.

Set the meter to volts DC, connect the red lead to the cable terminal at the starter motor and the ground to the battery negative terminal. Set in the seat and do all the things you're supposed to do to make the thing start. Hit the key switch and watch for a voltage reading on the meter or for your continuity light to, light. It does! You got a bad stater. It don't, move to next test.

Leave the ground connected and move your red test lead to the small wire terminal on the starter solenoid, the small terminal returning from the key switch. Hit the key switch. Got juice! The starter solenoid is bad. No reading. Then the key switch is bad or you have a failure in the safety switch system.
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There are so many variables in the safety interlock system that it's not really practical to try to get into every aspect in this format. Check all of the switch's to make sure they are operational. Make sure the plunger or lever mechanism moves and is being activated by its respective control. Flood the switch's with a carb or electrical spray cleaner in an effort to wash out dirt and grit. Some units have a control module or brain to sense if the engine is running or not. Especially when you have a sensor system that kills the ignition at start up. Obviouslly you don't want to kill the ignition when you engage the blade(s) to mow but some systems do kill the ignition if the blades are engaged when cranking the engine, some just prevent cranking. If yours has a module for some functions, it will be mounted most times to a grounded area for self grounding. It looks a lot like a signal flasher on a car but mounted to a mounting strap. It has several wires to it with one side normally an epoxied appearing surface.

Make sure this mounting is clean and tight. That's about all you can do with it. If that doesn't correct the problem and all the indiviadual switch's appear to be working right, suspect the key switch.

This is very important. Get a key switch with matching terminals, a switch made for lawn and garden equipment. Some auto supplies have them but make sure it is for lawn and garden equipment and not for automotive use or large tractors that use automotive type engines. Automotive ignition system use battery voltage and the switch turns on voltage to the ignition. Lawn and garden engine ignitions do not use battery voltage. Battery voltage will blow the igniton coil instantly. Lawn and garden equipment key switch's do not switch battery voltage to the ignition. When the key is turned off it grounds the ignition to shut the engine off, it never subjects the ignition coil to battery voltage.

The key switch should cost somewhere around 15 bucks. Not to bad. If that doesn't correct, then go back to that dern module and replace it. Depending on manufacturer they will run from $20 to $60, make it the last thing you replace, it's the least likely cause of the problem (except for the check for mounting being clean and tight) and the most expensive, so don't rush out and order one until you're sure about everything else.

Well, that about covers it without writting a book. Lots of variables, especially in the safety interlock area. Bottom line in that regard is don't stop checking or looking for things just because it isn't mentioned here. Some units are now installing a reverse kill switch to kill the engine when you put it in reverse, unless you first throw a reverse switch safety disable switch first. I know that sound nuts but it's true. Any other switch's tossed in there for what ever purpose can if not working right, be at the core of any problem. Get into a hurry and you're bound to miss something, so grab that third cup of coffee and get with it.