In epidemiology many exposures of interest are measured with error. Random, or 'classical', error in exposure measurements attenuates linear exposure-disease associations. However, its precise effects on different nonlinear associations are not well known. We use simulation studies to assess how classical measurement error affects observed association shapes and power to detect nonlinearity. We focus on a proportional hazards model for the exposure-disease association and consider six true association shapes of relevance in epidemiology: linear, threshold, U-shaped, J- shaped, increasing quadratic, asymptotic. The association shapes are modeled using three popular methods: grouped exposure analyses, fractional polynomials, P-splines. Under each true association shape and each method we illustrate the effects of classical exposure measurement error, considering varying degrees of random error. We also assess what we refer to as MacMahon's method for correcting for classical exposure measurement error under grouped exposure analyses, which uses replicate measurements to estimate usual exposure within observed exposure groups. The validity of this method for nonlinear associations has not previously been investigated. Under nonlinear exposure-disease associations, classical measurement error results in increasingly linear shapes and not always an attenuated association at a given exposure level. Fractional polynomials and P-splines give similar results and offer advantages over grouped exposure analyses by providing realistic models. P-splines offer greatest power to detect nonlinearity, however random exposure measurement error results in a potential considerable loss of power to detect nonlinearity under all methods. MacMahon's method performs well for quadratic associations, but does not in general recover nonlinear shapes.