A gravitational singularity is mathematically at the center of a black hole. There are thought to be many black holes in the universe, so to create a singularity that would swallow up the entire universe only requires the most massive black hole and time. A lot of time. As we understand it today, that would only require a black hole with a mass of 40,000,000,001 (yeah, forty-billion-and-one) solar masses — and far more time than the age of the Earth will ever be.
Can a nuclear bomb do that? No. Heck no.
Can one crazy world leader do it? No. Heck no.
Crafting a good story is about helping the reader to suspend their disbelief. You don't need to be right. You need to be believable enough. Therefore, let me introduce you to the Butterfly Effect, a concept within Chaos Theory that poetically suggests a butterfly in Texas can be the cause of a huricane in China.
Your problem will be figuring out a way to end the universe quickly. Quickly, as in soon enough to represent a real threat that the old lady next door would care about. Well, maybe not necessarily that soon, but you get my drift.
So, let's examine this a moment and see if it takes us anywhere. You don't want an explosion, so a nuclear weapon is useless. Why? Because to create a singularity you need to bring mass together, and explosions do just the opposite (ignoring the tendency for nukes to convert mass to energy, which doesn't help matters).
What you need is an implosion. Something that can draw a fearsome amount of mass together to form the largest singularity in the universe.
But, if you think about it, an implosion that could do that would destroy the universe just in the attempt to make the singularity that you you've been selling as the universe's coup d'jour. The implosion being much more threatening than the end result (and an embarrasing PR nightmare on top of that)... the end result is a confusing story. OK, so the implosion, which frankly would be a whomping difficult problem to explain in a believable manner in the first place, doesn't leave us with the intended bugaboo anyway.
So let's think about subatomic particles. They're all over the place. Let's say you can send waves out with subatomic particles. This might be believable enough that you could postulate for your story the use of subatomic particles to destroy the universe. No, you're not sucking everything into one unbelievably massive black hole. I've proven to myself that that particular plot element is too much to sell. What you're doing instead is setting a blanket that covers the universe on fire. And fires are started with matches. Suddenly, it's possible to come up with the butterfly you need to create your huricane. What might go "boom!" that would create enough disturbance in the sub-atomic ether such that the universe is sterilized?
I don't know... but I expect you'll have fun figuring it out!