It depends a lot on the geology of the mountain, of which I know nothing. Is the rock sedimentary? Does it have horizontal strata or do they slope steeply downwards? Is one of the strata weak?
It might be that all you have to do is shake it hard enough and gravity will do the rest. One really big hydrogen bomb? A medium or small one? Look up Cumbre Vieja (Canary islands) for a case of serious instability, and if you live on the USA Eastern seaboard, worry.
Somewhere on my travels, I forget exactly where, I passed a roadside memorial to the people who had once been camping in the valley below. There was no valley below, just a field of boulders. One summer day, after heavy rain which lubricated a fault, a mountain had collapsed downslope under its own weight. That's an extreme case but not a unique one.
It's said that the Swiss have permanent explosive mines in carefully selected locations in their mountain passes. If someone ever tries to invade, a large chunk of mountain will fall on them.