Negative energy? Consider the

Penrose process (Wikipedia).

So a particle traveling in a Kerr black hole's ergosphere can have negative energy as viewed from infinity.

That's an extreme case of something very common: negative potential energy.

Gravitational potential energy is always negative, and

electromagnetic potential energy is often negative.
Then there is the question of the total energy of the Universe. There is a big problem with that, because in general relativity, one cannot localize gravitational energy. One cannot find any gravitational energy-momentum tensor. One can find the total gravitational potential energy of some object, because one can observe its total mass from a distance. (Observed mass) - (constituents' mass) = (gravitational potential energy). One can help out by finding a gravitational energy-momentum pseudotensor and integrating it over volume. It's called a pseudotensor, because while it is indexed like a tensor, it does not have the appropriate coordinate-transformation properties. It's thus something like the connection coefficients.

The usual belief about the total energy of the Universe is that it is zero, that its constituents' mass and its gravitational-potential energy mass cancel out.

I think that the problem that some people have is that they think of energy as some sort of stuff, meaning that there cannot be a negative amount of it. But if energy is some quantitative attribute, like electric charge, then it can indeed be negative.