  We provide $L^p$-versus $L^\infty$-bounds for eigenfunctions on a real
spherical space $Z$ of wavefront type. It is shown that these bounds imply a
non-trivial error term estimate for lattice counting on $Z$. The paper also
serves as an introduction to geometric counting on spaces of the mentioned
type. Section 7 on higher rank is new and extends the result from v1 to higher
rank. Final version. To appear in Acta Math. Sinica.
