Let $f:\mathbb{R}^n \to \mathbb{R}^n$ be smooth. Does there always exist a hyperplane $V \subseteq \mathbb{R}^n$, such that $f(V)$ is contained in some hyperplane in the target?
Does the answer change if we only ask for $f(V)$ to be contained in some $n-1$ dimensional submanifold of $\mathbb{R}^n$?
Consider $$f_1(x,y)=(x\cos r^2+y\sin r^2, y\cos r^2-x\sin r^2)\qquad\text{with }r=\sqrt{x^2+y^2}$$ i.e., we rotate the more we move away from the origin. All lines (not only through the origin) get twisted, so are not ocontained in hyperplanes.
Next consider $$f_2(x,y)=f_1(x,y)\cdot \cos(r^2\sqrt 2).$$ Then the twirled lines even come back to the origin infinitely often and from irrationally spaced directions, i.e., from a dense set of directions, not fitting into a submanifold.