Re. "regular" and "singular" - is this in the parametric sense, or the geometric one? (i.e. is the 'singularity' a property of the curve, or just of its initial parametrization?)

Welcome to MathOverflow! Chapters 1 of Edmunds and Evans' [Spectral Theory and Differential operators][1] is (I think) easily the best reference for this kind of thing, and contains everything you could ever want to know about this :) [1]: scholar.google.co.uk/scholar_url?url=https://books.google.co‌​.uk/…

Have you looked in Federer's "Geometric Measure Theory"? I remember there being some discussion of this sort of thing shortly before the discussion of the Whitney extension theorem (which you may have come across already if you're thinking about things like this).

Forgive my ignorance but... can you explain how 'curvature' isn't an immediate impediment? (it's also possible/likely I'm just not understanding your question!)

(my $\nu_t$ is your $f(t,\cdot)$ by the way - I know I should really stick with your notation, but writing it in a more familiar way might help people (me) recognise certain things more easily)

This just might just be my general ignorance, but I'm struggling a bit to make sense of your question. It seems to me like you have a family $(\nu_t)_{t\in \mathbb{R}}$ of measures and want to know whether/when it's possible to make sense of the 'integral' $ \nu = \int_\mathbb{R}\nu_t\hspace{.2pc}\mathrm{d}t$ and, if so, whether $ \nu(E) = \int_\mathbb{R} \nu_t(E)\hspace{.2pc} \mathrm{d}t$ for $E\in \Sigma$. Is that anywhere close?

Just in case you're not aware of it already: there are lots of nice examples of this sort of thing in R. S. Hamilton's "The Inverse Function Theorem of Nash and Moser".

Isn't $F'(0)$ just the inclusion map sending $C^1([0,1])$ into $C([0,1])$? (or have I just displayed an embarrassing inability to differentiate ;))

I've always rather liked the "Pyramid Algorithms" approach to this stuff advocated by e.g. Ron Goldman. The Hermite-Genocchi/Kergin approach you mention leads to lots of interesting mathematics, but isn't something I think I'd have been able to appreciate fully as an undergraduate (though maybe that's not saying all that much!)

Have you thought about the case where the $T_n$ are 'approximation' operators? (e.g. $T_nx = (x_1,\dots,x_n,0,0,\dots)$ on $l^p$)

Re. everything being ok for the uniform norm: you're right and I was wrong. I didn't think it through - sorry! (I'm used to my composition operators being of the form $\Phi: f\mapsto f\circ \phi$)

Can you give any more background about what you want to do? e.g. why do you need $\Phi$ to be Lipschitz, how does the implicit function theorem fit in?

Re. I need an implicit function theorem (and just in case it helps) - there are also implicit function theorems for other types of topological vector spaces besides the Banach spaces (like the famous 'Nash-Moser' one)

$X = C_b(D,\mathbb{R}^n)$ and $X=C_b(\bar D, \mathbb{R}^n)$ are two 'boring' examples. The harmonic members (i.e. those which satisfy $\Delta f_i = 0$ for $i=1,...,n$) of these two spaces should also work (I think!), and might be a bit more interesting.

I think the problem here is more about norms than about spaces, as in 'once you have a norm which works, you can get lots of different spaces which do by taking completions of your favorite not-necessarily-closed linear subspaces of $C_b(D,\mathbb{R}^n)$'

I meant 'closed linear subspace'

Both of your non-examples involve spaces where the norm measures something related to difference quotients; I suspect you might have more luck with norms which measure other things. What about having $X$ be a closed subspace of $C_b(D,\mathbb{R}^n)$?

(and I'm certainly not suggesting that things which don't have obvious connections to applications aren't worthy of study)