  In this paper, we study the Cauchy problem for a quasilinear degenerate
parabolic stochastic partial differential equation driven by a cylindrical
Wiener process. In particular, we adapt the notion of kinetic formulation and
kinetic solution and develop a well-posedness theory that includes also an
$L^1$-contraction property. In comparison to the previous works of the authors
concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010)
1014-1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl.
123 (2013) 4294-4336], the present result contains two new ingredients that
provide simpler and more effective method of the proof: a generalized It\^{o}
formula that permits a rigorous derivation of the kinetic formulation even in
the case of weak solutions of certain nondegenerate approximations and a direct
proof of strong convergence of these approximations to the desired kinetic
solution of the degenerate problem.
