This paper considers hypothesis testing in semiparametric models which may be non-regular. I show that C($\alpha$) style tests are locally regular under mild conditions, including in cases where locally regular estimators do not exist, such as models which are (semi-parametrically) weakly identified. I characterise the appropriate limit experiment in which to study local (asymptotic) optimality of tests in the non-regular case, permitting the generalisation of classical power bounds to this case. I give conditions under which these power bounds are attained by the proposed C($\alpha$) style tests. The application of the theory to a single index model and an instrumental variables model is worked out in detail.

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