Could be a number of reason, such as bad spark plug, bad coil, ground wire damaged and shorting to ground, coil not getting connection to ground through mounting posts, or a safety switch problem if it's a rider.
Give me a bit more information and we'll work through this.
What type of equipment and model/serial numbers? What have you tried?
That's a classic machine you have there. It would have had a Tecumseh TNT100 engine on it. You've done the test with the spark plug out of the engine held against the block and saw no spark. Is that correct?
Can you tell me what distance you gapped the coil to the flywheel, and what you set the points at?
Does the flywheel have strong magnets if you try holding a screwdriver near them? It should pull a standard screwdriver dangled from about an inch.
Check these two things also, please.
Make sure the posts are free of corrosion and the coil is getting good contact to the posts by cleaning both surfaces with a bit of sandpaper.
Check the engine kill wire to make sure it's not damaged and grounding somwhere it shouldn't.
I'll be logged out for a bit and back on later to continue if this isn't your solution.