We survey recent work done on the values at integer points of irrational inhomogeneous quadratic forms, namely, inhomogeneous analogues of the famous Oppenheim conjecture. We also prove that the set of such forms in two variables whose set of values at integer points avoids a given countable set not containing zero has full Hausdorff dimension. Moreover, we consider the more refined variant of this problem, where the shift is fixed and the form is allowed to vary. The strategy is to translate the problems to homogeneous dynamics and deduce the theorems from their dynamical counterparts. While our approach is inspired by the work of Kleinbock and Weiss (Values of binary quadratic forms at integer points and Schmidt games. In: Recent Trends in Ergodic Theory and Dynamical Systems, Vadodara (2012). Contemporary Mathematics (Vol. 631, pp. 77–92). American Mathematical Society, 2015), the dynamical results can be deduced from more general results of An, Guan, and Kleinbock (Ergod Theory Dyn Syst 42(4):1327–1372).