In particular, I'm interested in the case we have a PL embedding $f : S^1 \to M$ (for a smooth manifold $M$): can it be homotoped to a smooth embedding? I'm not very familiar with PL stuff, so I may be misusing terminology; by PL embedding I mean a continuous map which is a homeomorphism onto its image, along with a triangulation of S^1 and a triangulation of M relative to which $f$ is linear on each simplex.
Evidently, the difficulty in this problem is to perform the smoothing of $f$ in a way as to have the final function be an immersion.
It would also be interesting to learn the answer of this in the case of a general PL embedding between two smooth manifolds.
In general, this is false. In particular, it is false already for embeddings $S^2 \to M^4$. For example take the manifold given by attaching a $2$-handle to a right-handed trefoil in $S^3=\partial B^4$ with framing $0$ (the interior if you insist on no boundary). There is a $PL$-embedding of $S^2$ given by taking the core disk of the $2$-handle and gluing it to the cone on the knot in $B^4$. However, there are relatively strong conditions on when a homology class in $H_2(M^4, \Bbb Z)$ can be represented by a smoothly embedded sphere and in this case the homology class has no such representative (by say the adjunction inequality for Stein surfaces).