In this article we prove path-by-path uniqueness in the sense of Davie [25] and Shaposhnikov [46] for SDE's driven by a fractional Brownian motion with a sufficiently small Hurst parameter H∈(0,12), uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.Using this result, we construct weak unique regular solutions in Wlock,p([0,1]×Rd), p>d of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lions [28], Ambrosio [2] or Crippa-De Lellis [23].Our approach is based on a priori estimates at the level of flows generated by a sequence of mollified vector fields, converging to the original vector field, and which are uniform with respect to the mollification parameter. In addition, we use a compactness criterion based on Malliavin calculus from [24] as well as a supremum estimate in time of moments of the derivative of the flow of SDE solutions.