  First, we present some results about the H\"older continuity of the sample
paths of so called dilatively stable processes which are certain infinitely
divisible processes having a more general scaling property than
self-similarity. As a corollary, we obtain that the most important
(H,delta)-dilatively stable limit processes (e.g., the LISOU and the LISCBI
processes, see Igloi [4]) almost surely have a local H\"older exponent H. Next
we prove that, under some slight regularity assumptions, any two dilatively
stable processes with stationary increments are singular (in the sense that
their distributions have disjoint supports) if their parameters H are
different. We also study the more general case of not having stationary
increments. Throughout the paper we specialize our results to some basic
dilatively stable processes such as the above-mentioned limit processes and the
fractional L\'evy process.
