Let $A_f$ be an abelian surface over $\mathbf{Q}$ of $\mathbf{GL}_2$-type arising from a weight $2$ cuspidal eigenform $f\in S_2(\Gamma_0(N))$. What is known (or expected to be true) for the size of the set of primes $p$ such that $A_f$ mod $p$ is ordinary (resp. non-ordinary)? Do you know of a reference where this is discussed?

anyabelian variety over a number field, the set of primes where the variety has ordinary reduction has density 1 "in the potential sense". As Felipe observed, this is a theorem in dimensions 1 and 2. The extension of the base is also conjectured to be the smallest field over which the monodromy groups are connected. – Barinder Banwait Nov 17 '12 at 20:45