5 edit number three.

When $M$ is a compact $2$-manifold, with or without boundary, $P(M)$ is known. When $M$ is a 3-manifold there's bits and pieces known, especially once you get to more fine detail like pseudo-isotopy embedding spaces. But at the level of $P(M)$ I don't think there's a complete description for a single $3$-manifold. For $4$-manifolds the situation is worse, but again there are some things known for pseudo-isotopy embedding spaces.

edit 2: Rick Litherland generalized a result of Zeeman (Deforming twist-spun knots TAMS 250 (1979) 311--331) showing that "Deform-spun knots" have complements that frequently fibre over $S^1$. This process called "Deform spinning" is just the boundary map in the pseudo-isotopy fibre sequence for spaces of knots. One of the nice things about this is Litherland gives a prescription for what the fibre is. In the case where you're looking at the pseudo-isotopy sequence for long knots in $\mathbb R^3$, it generates long embeddings of $\mathbb R^2$ in $\mathbb R^4$. So the fibre is a 3-manifold with boundary a sphere. This process produced some embeddings of 3-manifolds in the 4-sphere that nobody had known about at the time, like the once-punctured Poincare Dodecahedral Space (which without a puncture does not embed in $\mathbb R^4$, at least, not smoothly, it does admit a tame topological embedding). I got interested in this case largely because it represents sort of an extreme end of the terrain of your dissertation.

edit 3: I forgot to mention, I kept on pushing trying to understand why your dissertation broke down in co-dimension two. In some sense my paper "An obstruction to a knot being deform-spun via Alexander polynomials" is an answer. In a way it's not, since co-dimension two deform-spinning is more of a free loop space construction than a based loop space construction. But I found the exercise informative.

edit 2: Rick Litherland generalized a result of Zeeman (Deforming twist-spun knots TAMS 250 (1979) 311--331) showing that "Deform-spun knots" have complements that frequently fibre over $S^1$. This process called "Deform spinning" is just the boundary map in the pseudo-isotopy fibre sequence for spaces of knots. One of the nice things about this is Litherland gives a prescription for what the fibre is. In the case where you're looking at the pseudo-isotopy sequence for long knots in $\mathbb R^3$, it generates long embeddings of $\mathbb R^2$ in $\mathbb R^4$. So the fibre is a 3-manifold with boundary a sphere. This process produced some embeddings of 3-manifolds in the 4-sphere that nobody had known about at the time, like the once-punctured Poincare Dodecahedral Space (which without a puncture does not embed in $\mathbb R^4$, at least, not smoothly, it does admit a tame topological embedding). I got interested in this case largely because it represents sort of an extreme end of the terrain of your dissertation.

(a) Let $N$ be a co-dimension zero solid torus in $M=\mathbb R^3$ or $M=S^3$. There's the pseudo-isotopy embedding fibration $P(N,M) \to Emb(N,M)$. If $N$ is an unknotted solid torus, the question of what the map the image of the map $\pi_0 P(N,M) \to \pi_0 Emb(N,M)$ is, this is a long-standing hard problem in knot theory. Another way to say it is `which knots in $S^3$ bound a disc in $D^4$?'. These knots are called slice knots. Ralph Fox has the Slice Ribbon Conjecture, which might be described as a hopeful combinatorial answer to the question. There are many useful tools for determining whether or not a given knot is slice, recent slice,starting with the Alexander module and more recently tools from Heegaard Floer theoryhave been quite useful.
• Look at the class of knots that are a connect-sum of torus knots. Using the Alexander module Litherland proved such a knot bounds a disc in $D^4$ if and only if in the prime decomposition, the number of times a prime summand appears (like say a right-handed trefoil) is equal to the number of times its mirror-image appears (the left handed trefoil, in my example). Related previous MO thread
• Paolo Lisca has used Heegaard Floer theory to determine when a connect sum of 2-bridge knots in $S^3$ bounds a disc in $D^4$ REF, but in this case the answer is more elaborate.