We study competitive equilibrium in the canonical Fisher market model, but with indivisible goods. In this model, every agent has a budget of artificial currency with which to purchase bundles of goods. Equilibrium prices match between demand and supply---at such prices, all agents simultaneously get their favorite within-budget bundle, and the market clears. Unfortunately, a competitive equilibrium may not exist when the goods are indivisible, even in extremely simple markets such as two agents with exactly the same budget and a single item. Yet in this example, once the budgets are slightly perturbed---i.e., made generic---a competitive equilibrium is guaranteed to exist. In this paper we explore the extent to which generic budgets can guarantee equilibrium existence (and thus related fairness guarantees) in markets with multiple items. We complement our results in [Babaioff et al., 2019] for additive preferences by exploring the case of general monotone preferences, establishing positive results for small numbers of items and mapping the limits of our approach. We then consider cardinal preferences, define a hierarchy of such preference classes and establish relations among them, and for some classes prove equilibrium existence under generic budgets.