If it will not spark, pull the small ground wire from th base of the coil and try it. This will tell you whether it may be shorting to ground or if you have a bad coil.
You can test the switch by doing a continuity test with yout meter set to reads ohms. Look at the wires on the back of the switch. You should have continuity between the terminal going to ground and the terminal going to the engine coil when the switch is in the off position, and it should be open between these terminals in the on position.
You should have continuity between the wire coming from the battery and the one going to the small energizing wire on the side of the solenoid when the switch is in the start position.
Although this answers the specific questions you asked, I think there's more to this story, so please converse with me until we get your equipment going again.
When you reply can you please include the serial number of your equipment and the brand and model information of the engine so I can better assist?
Thank you!
Factory Authorized Trainer
Outdoor Power Equipment technical trainer since 1990, covering eight states.