We introduce a new methodology for analyzing serial data by quantile regression assuming that the underlying quantile function consists of constant segments. The procedure does not rely on any distributional assumption besides serial independence. It is based on a multiscale statistic, which allows to control the (finite sample) prob- ability for selecting the correct number of segments S at a given error level, which serves as a tuning parameter. For a proper choice of this parameter, this tends ex- ponentially fast to the true S, as sample size increases. We further show that the location and size of segments are estimated at minimax optimal rate (compared to a Gaussian setting) up to a log-factor. Thereby, our approach leads to (asymptotically) uniform confidence bands for the entire quantile regression function in a fully nonpara- metric setup. The procedure is efficiently implemented using dynamic programming techniques with double heap structures, and software is provided. Simulations and data examples from genetic sequencing and ion channel recordings confirm the ro- bustness of the proposed procedure, which at the same time reliably detects changes in quantiles from arbitrary distributions with precise statistical guarantees.