Let $E$ be an $n$-dimensional smooth manifold obtained as the total space of a fiber bundle with codimension-$m$ fibers, and let $X \subset E$ be a codimension-$k$ submanifold. (More generally, one could take $E$ to be any $n$-manifold with a codimension-$m$ foliation.)
Let $A \subset X$ be the set of points $x \in X$ where $X$ is tangent to the fiber containing $x$. Is there an appropriate notion of transversality for families of submanifolds that lets us understand $A$? After a small isotopy of $X$, can we assume that $A\subset X$ is a smooth submanifold, and can its (co)dimension be expressed in terms of $k$, $m$, and $n$?