I wonder whether it is possible to have a smooth action of the $ax+b$ Lie group with compactly supported fundamental vector fields on $\mathbb{R}^2$ in such a way that it is non-trivial at least at one point, i.e.linearily independent fundamtental vector fields or so.


The abelian case is possible (also in higher dimensions): there are $n$ commuting vector fields with compact support on $\mathbb{R}^n$ being linearily independent inside an open ball.
So §ax+b$ would be the first step into non-commutative examples. IN higher dimensions I have examples for say the Heisenberg group.



Action of $ax+b$ with compact support
No, it's not possible, because of the generalized Reeb Stability Theorem,
*A generalization of the Reeb Stability Theorem*, William P. Thurston, Topology, V13,
pp 347--352, 1974. The theorem basically says that for any group of $C^1$-smooth diffeomorphisms
of a manifold that has a fixed point where every element has first derivative trivial,
the group action near that point has a generalized nilpotent structure --- the
intersection of the lower central series is trivial, for some ordinal $\alpha$.
(My original interest was for understanding holonomy around leaves of foliations.)


This generalized nilpotence phenomenon is fairly obvious for anything detected by the
Taylor expansion at a point: if you look at the Taylor series for a vector field,
commutators of vector fields with trivial $0$th and first term vanish to an even higher
order. The main point is to understanding diffeomorphisms (or vector fields) that
either have
$C^\infity$ contact to the identity, or are not smooth enough to analyze with a Taylor
series. This phenomenon is also related to the phenomenon analyzed by Margulis and others,
that discrete groups of Lie groups generated by "small" elements are nilpotent.


For a Lie group, this result implies that any action near a fixed point where it has
$C^1$ contact to the identity factors through a nilpotent Lie group. For
a Lie group with a compactly supported action, apply this to a point on the frontier of any orbit,
to conclude that the orbit factors through the action of a nilpotent quotient (in particular
the group modulo the smallest term in its lower central series). For the affine group, this
quotient is $\mathbb R$.


