We consider shape-restricted nonparametric regression on a closed set $$\mathcal{X} \subset \mathbb{R},$$ where it is reasonable to assume that the function has no more than $$H$$ local extrema interior to $$\mathcal{X}$$. Following a Bayesian approach we develop a nonparametric prior over a novel class of local extremum splines. This approach is shown to be consistent when modelling any continuously differentiable function within the class considered, and we use itto develop methods for testing hypotheses on the shape of the curve. Sampling algorithms are developed, and the method is applied in simulation studies and data examples where the shape of the curve is of interest.