  The study of path behaviour of stochastic processes is a classical topic in
probability theory and related areas. In this frame, a natural question one can
address is: whether or not sample paths belong to a critical H\"older space?
The answer to this question is negative in the case of Brownian motion and many
other stochastic processes: it is well-known that despite the fact that
Brownian paths satisfy, on each compact interval $I$, a H\"older condition of
any order strictly less than $1/2$, they fail to belong to the critical
H\"older space $\mathcal{C}^{1/2}(I)$. In this article, we show that a
different phenomenon happens in the case of linear multifractional stable
motion (LMSM): for any given compact interval one can find a critical H\"older
space to which sample paths belong. Among other things, this result improves an
upper estimate, recently derived in Bierm\'e, Lacaux (2013), on behaviour of
LMSM, by showing that the logarithmic factor in it is not needed.
