  In this article we prove path-by-path uniqueness in the sense of Davie
\cite{Davie07} and Shaposhnikov \cite{Shaposhnikov16} for SDE's driven by a
fractional Brownian motion with a Hurst parameter $H\in(0,\frac{1}{2})$,
uniformly in the initial conditions, where the drift vector field is allowed to
be merely bounded and measurable.\par
  Using this result, we construct weak unique regular solutions in
$W_{loc}^{k,p}\left([0,1]\times\mathbb{R}^d\right)$, $p>d$ of the classical
transport and continuity equations with singular velocity fields perturbed
along fractional Brownian paths.\par
  The latter results provide a systematic way of producing examples of singular
velocity fields, which cannot be treated by the regularity theory of
DiPerna-Lyons \cite{DiPernaLions89}, Ambrosio \cite{Ambrosio04} or Crippa-De
Lellis \cite{CrippaDeLellis08}.\par
  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
\cite{DMN92} as well as supremum concentration inequalities.
  \emph{keywords}: Transport equation, Compactness criterion, Singular vector
fields, Regularization by noise.
