We use self-calibrating dispersion scan to experimentally detect and quantify the presence of pulse train instabilities in ultrashort laser pulse trains. We numerically test our approach against two different types of pulse instability, namely second-order phase fluctuations and random phase instability, where the introduction of an adequate metric enables univocally quantifying the amount of instability. The approach is experimentally demonstrated with a supercontinuum fibre laser, where we observe and identify pulse train instabilities due to nonlinear propagation effects under anomalous dispersion conditions in the photonic crystal fibre used for spectral broadening. By replacing the latter with an all-normal dispersion fibre, we effectively correct the pulse train instability and increase the bandwidth of the generated coherent spectrum. This is further confirmed by temporal compression and measurement of the output pulses down to 15 fs using dispersion scan.