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When can the distributional assumptions of theorems and learning algorithms be trusted? Inspired by this question, Rubinfeld and Vasilyan (2023) initiated the study of testable learning. In this schema, we always learn one of the following two things: either we have achieved the desired accuracy regardless of whether the distributional assumptions are satisfied, or the input distribution does not satisfy the original distributional assumptions. Motivated by the challenge of relying on strong distributional assumptions in many theorems in mechanism design, we develop a testable learning framework for mechanism design. Traditional models in mechanism design assume that value distributions satisfy some notion of regularity. Unfortunately, testing regularity is not possible in the original testable learning framework as we show. To bypass this impossibility, we propose a regularized version of the testable learning framework. Under this framework, we always learn one of the following two things: either we achieve high revenue compared to the best possible revenue of any regular distribution close to the input distribution, or the input distribution does not satisfy regularity. We then use this framework to provide: 1) a tester-learner pair for revenue optimal mechanisms, 2) a tester for whether the fundamental Bulow-Klemperer Theorem (Bulow and Klemperer 1996) is applicable to a given dataset, and 3) a tester to confirm the existence of an anonymous reserve price that results in the anonymous price auction securing a constant fraction of the optimal revenue.