  We analyze thermonuclear and pycnonuclear fusion reactions in dense matter
containing atomic nuclei of different types. We extend a phenomenological
expression for the reaction rate, proposed recently by Gasques et al. (2005)
for the one-component plasma of nuclei, to the multi-component plasma. The
expression contains several fit parameters which we adjust to reproduce the
best microscopic calculations available in the literature. Furthermore, we show
that pycnonuclear burning is drastically affected by an (unknown) structure of
the multi-component matter (a regular lattice, a uniform mix, etc.). We apply
the results to study nuclear burning in a carbon_12-oxygen_16 mixture. In this
context we present new calculations of the astrophysical S-factors for
carbon-oxygen and oxygen-oxygen fusion reactions. We show that the presence of
a CO lattice can strongly suppress carbon ignition in white dwarf cores and
neutron star crusts at densities > 3e9 g cm^{-3} and temperatures T<1e8 K.
