MG 03 Rosso Corsa. With engine run button in the "run" position, starter will not engage when start button is pushed, n o clicking, nothing. Battery good lights bright, horn loud. Pushing run button in and out causes a small clicking sound under the seat somewhere around all the relays.

When you turn the key to start ,what happens ? The starter does or dosen't engage an spin the engine ?

I turn key it's says nothing ????? What do you want it to say ????
The way to a fix is to learn how it works an test the electrical circuit's using a DVOM - digital volt ohm meter . Viewing a wiring diagram to see what is involved in the starter system . Free wiring diagrams here http://www.bbbind.com/free-tsb Enter vehicle info. year , make , model an engine size. Under system click on engine , then under subsystem click on starting .Click the search button then the blue link.
Go on youtube an learn how to test electrical circuit's .

First, check the "Run/Stop" switch on the right handlebar switch housing. Make sure it's in the "Run" position. If it is, your starter relay may not be working. The relay is exactly like those on your car, a one inch plastic cube. Now, where it's located, I'm not sure. It could be under the seat or behind either of the two side covers on your bike. Once you find it, turn the switch on, hold on to the relay, and push the start button. You should feel the relay click. If it does click but the starter does not engage, you need to test the voltage coming out of the relay.

To test the output of the relay, look at the end of your starter from the right side of your bike. You'll see a five sided plate on the end of the starter solenoid. At the top of the plate, you'll see a single wire plugged into the starter. Unplug the wire and stick a test light probe in the connector on the end of the wire. Again, turn the switch on and press the start button. If the light lights up, your relay is good. If not, the relay is probably bad.

To do a final test on the starter, remove the five sided plate by removing the three small bolts that secure it. You'll see a plunger inside the solenoid. Make sure the bike's transmission is in neutral. Using a large screwdriver, push the plunger into the starter solenoid. The starter should engage and turn the engine over regardless of whether the switch is "on" or "off". Make certain that you push the plunger all the way in. If the starter does not turn, you starter is bad.

Good Luck
Steve

Your starter could be jamed causing the fly wheel to lock up try pushing the car in reverse let out the clutch with gear in 2nd forward position this will turn the engine in the opposite direction and free the starter if it is an automatic the starter needs to be removed this is caused by a rusted Bendix spring have the starter servised