  Let $d$ and $n$ be integers satisfying $C\leq d\leq \exp(c\sqrt{\ln n})$ for
some universal constants $c, C>0$, and let $z\in \mathbb{C}$. Denote by $M$ the
adjacency matrix of a random $d$-regular directed graph on $n$ vertices. In
this paper, we study the structure of the kernel of submatrices of $M-z\,{\rm
Id}$, formed by removing a subset of rows. We show that with large probability
the kernel consists of two non-intersecting types of vectors, which we call
very steep and gradual with many levels. As a corollary, we show, in
particular, that every eigenvector of $M$, except for constant multiples of
$(1,1,\dots,1)$, possesses a weak delocalization property: its level sets have
cardinality less than $Cn\ln^2 d/\ln n$. For a large constant $d$ this provides
a principally new structural information on eigenvectors, implying that the
number of their level sets grows to infinity with $n$. As a key technical
ingredient of our proofs we introduce a decomposition of $\mathbb{C}^n$ into
vectors of different degrees of `structuredness', which is an alternative to
the decomposition based on the least common denominator in the regime when the
underlying random matrix is very sparse.
