We study classes of set partitions determined by the avoidance of multiple patterns, applying a natural notion of partition containment that has been introduced by Sagan. We say that two sets S and T of patterns are equivalent if for each n the number of partitions of size n avoiding all the members of S is the same as the number of those that avoid all the members of T.

Our goal is to classify the equivalence classes among two-element pattern sets of several general types. First, we focus on pairs of patterns {sigma, tau}, where sigma is a pattern of size three with at least two distinct symbols and tau is an arbitrary pattern of size k that avoids sigma.

We show that pattern-pairs of this type determine a small number of equivalence classes; in particular, the classes have on average exponential size in k. We provide a (sub-exponential) upper bound for the number of equivalence classes, and provide an explicit formula for the generating function of all such avoidance classes, showing that in all cases this generating function is rational.

Next, we study partitions avoiding a pair of patterns of the form (1212, tau), where tau is an arbitrary pattern. Note that partitions avoiding 1212 are exactly the non-crossing partitions.

We provide several general equivalence criteria for pattern-pairs of this type, and show that these criteria account for all the equivalences observed when tau has size at most six. In the last part of the paper, we perform a full classification of the equivalence classes of all the pairs {sigma, tau}, where sigma and tau have size four.