  We study the motivic Serre invariant of a smoothly bounded algebraic or rigid
variety $X$ over a complete discretely valued field $K$ with perfect residue
field $k$. If $K$ has characteristic zero, we extend the definition to
arbitrary $K$-varieties using Bittner's presentation of the Grothendieck ring
and a process of N\'eron smoothening of pairs of varieties.
  The motivic Serre invariant can be considered as a measure for the set of
unramified points on $X$. Under certain tameness conditions, it admits a
cohomological interpretation by means of a trace formula. In the curve case, we
use T. Saito's geometric criterion for cohomological tameness to obtain more
detailed results. We discuss some applications to Weil-Ch\^atelet groups, Chow
motives, and the structure of the Grothendieck ring.
