We prove that a homogeneous and isotropic universe containing a scalar field with a power-law potential, $V(��)=A��^{n}$, with $00$ 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 exist for non-integer $n>1$. More precisely, if $k

5 pages. v2: typos corrected, references added and some sections extended. Accepted Phys. Rev. D