Summary: We study a generalized class of supersolutions, so-called \(p\)-supercaloric functions, to the parabolic \(p\)-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for \(p\geq 2\), but little is known in the fast diffusion case \(1<p<2\). Every bounded \(p\)-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic \(p\)-Laplace equation for the entire range \(1<p<\infty\). Our main result shows that unbounded \(p\)-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case \(\frac{2n}{n+1}<p<2\). The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case \(1<p\leq \frac{2n}{n+1}\) and the theory is not yet well understood.