  Given a negatively curved geodesic metric space $M$, we study the statistical
asymptotic penetration behavior of (locally) geodesic lines of $M$ in small
neighborhoods of points, of closed geodesics, and of other compact (locally)
convex subsets of $M$. We prove Khintchine-type and logarithme law-type results
for the spiraling of geodesic lines around these objets. As a consequence in
the tree setting, we obtain Diophantine approximation results of elements of
non-archimedian local fields by quadratic irrational ones.
