Summary: Lagrangian coherent structures (LCSs) are time-varying entities which capture the most influential transport features of a flow. These can for example identify groups of particles which have greatest stretching, or which maintain a coherent jet or vortical structure. While many different LCS methods have been developed, the impact of the inevitable measurement uncertainty in realistic Eulerian velocity data has not been studied in detail. This article systematically addresses whether LCS methods are self-consistent in their conclusions under such uncertainty for nine different methods: the finite time Lyapunov exponent, hyperbolic variational LCSs, Lagrangian averaged vorticity deviation, Lagrangian descriptors, stochastic sensitivity, the transfer operator, the dynamic Laplacian operator, fuzzy c-means clustering and coherent structure colouring. The investigations are performed for two different realistic data sets: a computational fluid dynamics simulation of a Kelvin-Helmholtz instability, and oceanographic data of the Gulf Stream region. Using statistics gleaned from stochastic simulations, it is shown that the methods which detect full-dimensional coherent flow regions are significantly more robust than methods which detect lower-dimensional flow barriers. Additional insights into which aspects of each method are self-consistent, and which are not, are provided.