  We confront two integrability criteria for rational mappings. The first is
the singularity confinement based on the requirement that every singularity,
spontaneously appearing during the iteration of a mapping, disappear after some
steps. The second recently proposed is the algebraic entropy criterion
associated to the growth of the degree of the iterates. The algebraic entropy
results confirm the previous findings of singularity confinement on discrete
Painlev\'e equations. The degree-growth methods are also applied to
linearisable systems. The result is that systems integrable through
linearisation have a slower growth than systems integrable through isospectral
methods. This may provide a valuable detector of not just integrability but
also of the precise integration method. We propose an extension of the Gambier
mapping in $N$ dimensions. Finally a dual strategy for the investigation of the
integrability of discrete systems is proposed based on both singularity
confinement and the low growth requirement.
