  Let $E$and $F$ be Hermitian vector bundles over a complex manifold $X$ and
let $g\colon E\to F$ be a holomorphic morphism. We prove a Poincar\'e-Lelong
type formula with a residue term $M^g$. The currents $M^g$ so obtained have an
expected functorial property. We discuss various applications: If $F$ has a
trivial holomorphic subbundle of rank $r$ outside the analytic set $Z$, then we
get currents with support on $Z$ that represent the Bott-Chern classes $\hat
c_k(E)$ for $k >\rank E-r$. We also consider Segre and Chern forms associated
with certain singular metrics on $F$. The multiplicities (Lelong numbers) of
the various components of $M^g$ only depend on the cokernel of the adjoint
sheaf morphism $g^*$. This leads to a notion of distinguished varieties and
Segre numbers of an arbitrary coherent sheaf, generalizing these notions, in
particular the Hilbert-Samuel multiplicity, in case of an ideal sheaf.
