In the classical dimension (knots in $S^3$), it is considered standard (I think?) that the following sets are in bijective correspondence:

- locally flat knots up to ambient isotopy;
- PL-knots up to PL ambient isotopy;
- smooth knots up to smooth ambient isotopy.

This should be "by work of Moise" but I am having trouble finding references.

The textbooks of Burde-Zieschang (Proposition 1.10) and Kawauchi (Appendix A) seem to indicate that the set of PL-knots up to ambient isotopy coincides with the set of PL-knots up to PL-ambient isotopy. Some authors also work with tame knots which are defined as knots that are ambient isotopic to a PL-knot. So, it seems that the previous two sets also agree with the set of tame knots up to either ambient or PL ambient isotopy.

Does someone know where I can find the statements that involve the locally flat and smooth cases?

The closest I could get was 1.11.6 and 1.11.7 of Cromwell's book that seems to sketch a proof that the set of PL-knots up to isotopy coincides with the set of smooth knots up to ambient isotopy (the category mixing is unfortunate). Most textbooks appear to stick to the PL knots.

Related but distinct MO posts include: Reference for a fact (?) on homeomorphic knot complements as well as Reference request: A knot is tame if and only if it has a tubular neighbourhood