We study singular Hermitian metrics on vector bundles. There are two main
results in this paper. The first one is on the coherence of the higher rank analogue of multiplier
ideals for singular Hermitian metrics defined by global sections. As an application, we show
the coherence of the multiplier ideal of some positively curved singular Hermitian metrics whose
standard approximations are not Nakano semipositive. The aim of the second main result is to
determine all negatively curved singular Hermitian metrics on certain type of vector bundles, for
example, certain rank 2 bundles on elliptic curves.