What techniques can be used to prove that a spacetime is not asymptotically flat? The modern coordinate-indepenent definition of asymptotic flatness was introduced by Geroch in 1972. You can find presentations in Wald 1984 and Townsend 1997. The definition is in terms of existence of something  -- a certain type of conformal compactification -- and in many cases such as the Schwarzschild metric, it's straightforward to think up a construction that proves existence. But to prove that a spacetime is not asymptotically flat, you have to prove the nonexistence of any such compactification, which seems harder.
What techniques could be used to prove that a spacetime is not asymptotically flat?
The only example of such a technique that I've been able to think of is the following. Suppose we want to prove that an FLRW spacetime is not asymptotically flat. Wald gives a theorem by Ashtekar and Hansen that says that if a spacelike submanifold contains $i^0$, it admits a metric that is nearly Euclidean, so that, e.g., the spatial Ricci tensor falls off like $O(1/r^3)$. This implies that there can't be any lower bound on the Ricci tensor, but an FLRW spacetime can have such a lower bound on a constant-time slice, since such a slice has constant spatial curvature.
Townsend, http://arxiv.org/abs/gr-qc/9707012
Wald, General Relativity
 A: The notion of asymptotic flatness in 4-dimensions was studied way back in 1962 by Bondi, van der Burg, Metzner (here) and Sachs (here) and more recently by Barnich and Troessaert (the first few papers here)
They described asymptotic flatness in terms of the Bondi coordinates, where the metric takes the form
$$
ds^2 = \frac{V}{r} e^{2\beta} du^2 - 2 e^{2\beta} du dr + g_{AB} \left( dx^A - U^A du \right) \left( dx^B - U^B du \right)
$$
where $A,B = 2,3$, $x^A = (\theta,\phi)$ and $\det g_{AB} = r^4 \sin^2\theta$. Bondi shows that every 4-dimensional metric can be written in the form above. Techniques developed later by Penrose showed that one should really set a more general condition wherein $\det g_{AB} = \frac{1}{4} r^4 e^{2 {\tilde\varphi}}$ (following Barnich's notation). An asymptotically flat spacetime then satisfies these boundary conditions (at large $r$)


*

*$g_{AB} = r^2 \gamma_{AB} + {\cal O}(r)$ where $\gamma_{AB}$ is conformal to the metric of $S^2$, i.e.
$$
\gamma_{AB}dx^A dx^B = e^{2\phi(u,\theta,\phi)} \left( d\theta^2 + \sin^2\theta d\phi^2 \right)
$$

*$\frac{V}{r} = - 2 r \partial_u {\tilde \varphi} + \Delta {\tilde \varphi} + {\cal O}(r^{-1})$

*$\beta = {\cal O}(r^{-2})$

*$U^A = {\cal O}(r^{-2})$
Given a metric, its asymptotic flatness can be checked by testing if the above boundary conditions hold. 
EDIT: This is a discussion of asymptotic flatness at $\mathscr{I}^+$. An analogous discussion exists for $\mathscr{I}^-$ (simply take $u \to v = u + 2 r$. For a complete description of asymptotic flatness, one must also consider the structure of the spacetime at $i^0$. While this has been discussed in Ashtekar, Hansen, I do not know too much about it. I will leave this to other members to discuss.
