  We study the effects of noise cross-correlations on the steady states of
driven, nonequilibrium systems, which are described by two stochastically
driven dynamical variables, in one dimension. We use a well-known
stochastically driven coupled model with two dynamical variables, where one of
the variables is autonomous being independent of the other, whereas the second
one depends explicitly on the former. Introducing cross-correlations of the two
noises in the two dynamical equations, we show that depending upon the details
of the nonlinear coupling between the dynamical fields, such cross-correlations
can induce instabilities in the models, that are otherwise stable in the
absence of any cross-correlations. { We argue that this is reminiscent of the
roughening transition found in the Kardar-Parisi-Zhang equation in dimensions
greater than two.} Phenomenological implications of our results are discussed.
