This is more a comment than an answer but it's too long for a comment.
In ergodic theory (as opposed to dynamical systems), systems in which the acting group is not the integers or the reals have been widely studied, both for their own intrinsic interest and because of deep and striking applications to number theory and other areas. For example, the proof by Einsiedler, Katok, Lindenstrauss that the set of exceptions to Littlewood's conjecture has zero Hausdorff dimension uses an ergodic-theoretic result on higher rank group actions.
As soon as the acting group has higher rank (even in the simplest case, i.e. $\mathbb{Z}^2$), the study of their ergodic theoretic properties becomes dramatically more complicated, even if the phase space is as simple as possible (the circle). A famous example is Furstenberg's $\times 2\times 3$ problem: what are the measures which are simultaneously invariant under $x\to 2x \bmod 1$ and $x\to 3x\bmod 1$? (this corresponds to the action of $\mathbb{N}^2$ on $[0,1]$ given by $(a,b)\cdot x=2^a 3^b x\bmod 1$). It is suspected that there are very few invariant measures, which illustrates a general (conjectured or proved) phenomenon: higher rank dynamical systems tend to have few invariant measures, all of them with some algebraic structure. Indeed, Einsiedler, Katok and Lindenstrauss use such a rigidity result.
The book by Einsiedler and Ward "Ergodic thory (with a view towards number theory)" is an excellent reference for this general topic.
Unfortunately you'll have to face a «no» answer, as there is no general/generic way to handle your problem…
First of all, your definition of «non-hyperbolic» is not clear to me when $n>1$. As far as I know a singular point like yours is deemed hyperbolic when $0$ does not belong to the real convex hull of the spectrum (in $\mathbb C$) of the linear part $A$. In case the singularity is hyperbolic, Poincaré's linearization theorem ensures that the system is dynamically equivalent to its linear part, as you refer to in the question. Certainly when you have null eigenvalues the system is not hyperbolic in that sense.
Now if the singular point is not hyperbolic, then a lot of bad behaviors can crop up, even in the case $n=2$. The presence of resonances ($\mathbb Q$-linear relations between the eigenvalues of $A$) or quasi-resonances (irrational relations which are «well approximated» by rational ones) can mess things up. Even from a formal viewpoint the answer is not clear, and certainly not computable without restrictive conditions on $f$ (e.g. polynomial, and even then…). In particular it is not true that the system $\dot x=f(x)$ is conjugate (dynamically equivalent) to a system where $f$ is algebraic using changes of coordinates which are merely $C^1$ at the singular point. Hence it is not sufficient to take a finite jet (let alone the second jet) to answer your question when there are resonances (as is the case here when at least one eigenvalue vanishes).
Dynamical systems with at least one zero eigenvalue are called «saddle-nodes», so you might wish to search for this keyword. Notice, though, that the theory of saddle-nodes is complete only in the smooth case (that is, $f$ smooth) for $n=2$, and partially covered when $n>2$. The more resonances (e.g. zero eigenvalues) the more difficult the problem is, even from a purely theoretical viewpoint…
So, without more information on your specific system, nothing can be said.
EDIT: You still can say something regarding stability. For instance look at the system $$\dot x(t) = x(t)^{k+1} \\ \dot y(t)=-y(t)$$ with $x, y\in\mathbb R$. Depending on the parity of $k$, either all trajectories are stable or only a single trafectory is (that's $y=0$). In these problems you need to identify directions in the $x$-variable where you have contributions of exponential terms $\exp(x^{-k}/k)$. This can be done by inspecting the linear part, and in general governs «how stable» the solutions will be. I don't have a recipe when $n>1$, though.
Best Answer
A good reference for this sort of thing is Guckenheimer and Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, 1983, if you're not already familiar with it (and even if you are, for that matter). Chapter 3 in particular is relevant to your question.