We study the non-parametric estimation of an unknown survival function S with support on R+ based on a sample with multiplicative measurement errors. The proposed fully-data driven procedure is based on the estimation of the Mellin transform of the survival function S and a regularisation of the inverse of the Mellin transform by a spectral cut-off. The upcoming bias-variance trade-off is dealt with by a data-driven choice of the cut-off parameter. In order to discuss the bias term, we consider the Mellin-Sobolev spaces which characterize the regularity of the unknown survival function S through the decay of its Mellin transform. For the analysis of the variance term, we consider the i.i.d. case and incorporate dependent observations in form of Bernoulli shift processes and beta mixing sequences. Additionally, we show minimax-optimality over Mellin-Sobolev spaces of the spectral cut-off estimator.