We prove that a homogeneous and isotropic universe containing a scalar field with a power-law potential, V(ϕ)=Aϕ^n, with 0<n<1 and A>0 always develops a finite-time singularity at which the Hubble rate and its first derivative are finite, but its second derivative diverges. These are the first examples of cosmological models with realistic matter sources that possess weak singularities of “sudden” type. We also show that a large class of models with even weaker singularities exists for noninteger n>1. More precisely, if k<n<k+1 where k is a positive integer then the first divergence of the Hubble rate occurs with its (k+2)th derivative. At early times these models behave like standard large-field inflation models but they encounter a singular end state when inflation ends. We term this singular inflation.