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The category of flat vector bundles is equivalent to the category of local systems, see for instance this MO-question, which in turn are equivalent to representations of the fundamental group, see this MO-question. Via these correspondences, the vector bundle associated to the covering corresponds to the representation of the fundamental group of $X$ on the fibre of $f:Y\to X$ via deck transformations. In particular, the vector bundles associated to coverings are flat. A flat vector bundle has trivial characteristic classes, but triviality of characteristic classes is not sufficient for flatness, as the answer of abx shows.