Among the foundational results in differential topology are the <a href="http://en.wikipedia.org/wiki/Morse_function#The_Morse_lemma">Morse lemmas</a>:
<ol>
<li>Suppose that $f\colon\, M\to \mathbb{R}$ is a smooth function on a closed manifold M, that a is less than b, that $f^{−1}[a,b]$ is compact, and that there are no critical values between $a$ and $b$. Then $M^a:=f^{-1}(-\infty,a]$ is diffeomorphic to $M_b$.</li>
<li>
Let $f\colon\, M\to \mathbb{R}$ be a smooth function on a closed manifold M, with no critical points on $f^{-1}[-\epsilon,\epsilon]$ except k nondegenerate ones on $f^{-1}(0)$, all of index $s$. Then $f^{-1}[-\infty,\epsilon]$ is diffeomorphic to $X(f^{-1}[-\infty,-\epsilon];f_1,\ldots,f_k;s)$ (for suitable f<sub>i</sub>).<br>
Here $X(M;f;s)$ for $f\colon\,(\partial D^s)\times D^{n-s}\to M$ is M with an s-handle attached by f.
</li>
</ol>
One of the things that makes me feel that I don't understand Morse's lemmas as well as I would like to is that the conditions on the source and target of the Morse function f seem unnecessarily restrictive. Maybe we'd like M to be a manifold with boundary or with corners, or a stratified space, or an infinite-dimensional something-or-other? Indeed, analogues of the Morse lemmas continue to hold (but how much CAN we relax our requirements on M?).<br><br> 
On the target side, what about if we want the target to be something other than $\mathbb{R}$? <a href="http://en.wikipedia.org/wiki/Circle-valued_Morse_theory">Circle-valued Morse theory</a> and <a href="http://www.pnas.org/content/108/20/8122.full">Morse 2-functions</a> deal with Morse functions to S<sup>1</sup> and to R<sup>2</sup> correspondingly, and are quite useful.<br><br>
And so, in order to feel I have a bit more of a grip on the meta-mathematical conceptual framework of the Morse Lemmas, I'm very much interested in the following question:
<blockquote>
Let $f\colon\, M\to N$ be a smooth function, with non-degenerate critical points (critical points which don't vanish after a small perturbation). What conditions do I have to impose on M and on N to obtain reasonable analogues of the Morse Lemmas?
</blockquote>
By <i>reasonable analogues</a>, I mean lemmas which explicitly relate $f^{-1}(N^{\prime})$ to $f^{-1}(N^{\prime\prime})$ up to diffeomorphism (by some sort of generalized handle attachment operation?), where $N^{\prime}\subseteq N^{\prime\prime}\subseteq N$ are some reasonable analogues in N to $(-\infty,p]$ and $(-\infty,p+\epsilon]$ correspondingly.<br><br>
Is there any work, or any results along these lines? Is this all well-known (and easy?), is it open, or is it known to be impossible (as in: "if the target isn't R then something goes disastrously wrong")?