  We consider discrete random Schr\"odinger operators on $\ell^2
(\mathbb{Z}^d)$ with a potential of discrete alloy-type structure. That is, the
potential at lattice site $x \in \mathbb{Z}^d$ is given by a linear combination
of independent identically distributed random variables, possibly with
sign-changing coefficients. In a first part we show that the discrete
alloy-type model is not uniformly $\tau$-H\"older continuous, a frequently used
condition in the literature of Anderson-type models with general random
potentials. In a second part we review recent results on regularity properties
of spectral data and localization properties for the discrete alloy-type model.
