We provide a series of partial negative answers to the question raised in [Coron, Contemp. Math 2007] concerning the controllability, in the framework of L^2 solutions, of the viscous Burgers equation u_t + (u 2 /2)_x = u_xx for initial and terminal data prescribed for x ∈ (0, 1). We adapt the scaling and compactness technique of [Andreianov, Ghoshal, Koumatos, Part I] conceived for L^∞ solutions and based upon non-controllability, by Kruzhkov entropy solutions, of the inviscid Burgers equation. To this end, we develop a basic well-posedness theory of unbounded entropy solutions to the Cauchy problem for multi-dimensional scalar conservation laws with pure L^p data and polynomial growth up to the critical power p of the flux function. The case of the Cauchy-Dirichlet problem for the Burgers equation on an interval is also addressed, in the L^2 solution framework which is considerably simpler than the established L^1 theory of renormalized solutions to such problems. Local regularity of unbounded entropy solutions is discussed in the one-dimensional case with convex flux. We deduce non-controllability results for the viscous Burgers equation under the bounded amplification restriction of the form u 2 ≤ LT |C| and an additional L^2 − L^3 loc regularization assumption.