According to our current understanding of general relativity, the curvature of spacetime at a singularity is thought to be infinite. However, it's important to note that singularities are points where our current theories break down and our understanding of physics becomes limited. Therefore, the concept of infinity may not be applicable in the conventional sense.
A singularity is a region where the curvature of spacetime becomes infinitely strong, typically associated with black holes or the Big Bang. In the case of a black hole, for example, the singularity is located at the center, where the mass is compressed to a point of infinite density. At this point, the gravitational forces become extremely strong, and the equations of general relativity yield infinities.
However, these infinities could also be an indication that our understanding of the physics involved is incomplete. It's possible that a theory of quantum gravity, which combines the principles of quantum mechanics with general relativity, would provide a more complete description of what happens at singularities and resolve the infinities. Unfortunately, a complete theory of quantum gravity is currently elusive, and more research is needed to gain a better understanding of these extreme conditions.