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A set $\mathcal{A}\subset C[0,1]$ is \emph{shy} or \emph{Haar null } (in thesense of Christensen) if there exists a Borel set $\mathcal{B}\subset C[0,1]$and a Borel probability measure $\mu$ on $C[0,1]$ such that $\mathcal{A}\subset\mathcal{B}$ and $\mu\left(\mathcal{B}+f\right) = 0$ for all $f \in C[0,1]$.The complement of a shy set is called a \emph{prevalent} set. We say that a setis \emph{Haar ambivalent} if it is neither shy nor prevalent.  The main goal of the paper is to answer the following question: What can wesay about the topological properties of the level sets of the prevalent/non-shymany $f\in C[0,1]$?  The classical Bruckner--Garg Theorem characterizes the level sets of thegeneric (in the sense of Baire category) $f\in C[0,1]$ from the topologicalpoint of view. We prove that the functions $f\in C[0,1]$ for which the samecharacterization holds form a Haar ambivalent set.  In an earlier paper we proved that the functions $f\in C[0,1]$ for whichpositively many level sets with respect to the Lebesgue measure $\lambda$ aresingletons form a non-shy set in $C[0,1]$. The above result yields that thisset is actually Haar ambivalent. Now we prove that the functions $f\in C[0,1]$for which positively many level sets with respect to the occupation measure$\lambda\circ f^{-1}$ are not perfect form a Haar ambivalent set in $C[0,1]$.  We show that for the prevalent $f\in C[0,1]$ for the generic $y\in f([0,1])$the level set $f^{-1}(y)$ is perfect.  Finally, we answer a question of Darji and White by showing that the set offunctions $f \in C[0,1]$ for which there exists a perfect $P_f\subset [0,1]$such that $f'(x) = \infty$ for all $x \in P_f$ is Haar ambivalent.

Bruckner--Garg-type results with respect to Haar null sets in $C[0,1]$