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Ad-hoc constraints (also called generic constraints) are important for modelling Constraint Satisfaction Problems (CSPs). Many representations have been proposed to define ad-hoc constraints, such as tables, decision diagrams, binary constraint trees, automata and context-free grammars. However, prior works mainly focus on efficient Generalized Arc Consistency (GAC) propagators of ad-hoc constraints using the representations. In this paper, we ask a more fundamental question which bears on modelling constraints in a CSP as ad-hoc constraints, how the choice of constraints and operations affect tractability. Rather than ad-hoc constraints and their GAC propagators, our focus is on their expressive power in terms of succinctness (polysize) and cost of operations/queries (polytime). We use a large set of constraint families to investigate the expressive power of 14 existing ad-hoc constraints. We show a complete map of the succinctness of the ad-hoc constraints. We also present results on the tractability of applying various operations and queries on the ad-hoc constraints. Finally, we give case studies illustrating how our results can be useful for questions in the modelling of CSPs.