  The Kurdyka-{\L}ojasiewicz (K{\L}) property, exponent and modulus have played
a very important role in the study of global convergence and rate of
convergence for optimal algorithms. In this paper, at a stationary point of a
locally lower semicontinuous function, we obtain complete characterizations of
the K{\L} property and the K{\L} modulus via the outer limiting subdifferential
of an auxilliary function and a newly-introduced subderivative function
respectively. In particular, for a class of prox-regular, twice
epi-differentiable and subdifferentially continuous functions, we show that the
K{\L} property and the K{\L} modulus can be described by its Moreau envelopes
and a quadratic growth condition. We apply the obtained results to establish
the K{\L} property with exponent $\frac12$ and to provide calculation of the
modulus for a smooth function, the pointwise maximum of finitely many smooth
functions and regularized functions respectively. These functions often appear
in the modelling of structured optimization problems.
