  The Dressian Dr(k,n) parametrizes all tropical linear spaces, and it carries
a natural fan structure as a subfan of the secondaryfan of the hypersimplex
\Delta(k,n). We explore the combinatorics of the rays of Dr(k,n), that is, the
most degenerate tropical planes, for arbitrary k and n. This is related to a
new rigidity concept for configurations of n-k points in the tropical
(k-1)-torus. Additional conditions are given for k=3. On the way, we compute
the entire fan Dr(3,8).
