  Studying sample path behaviour of stochastic fields/processes is a classical
research topic in probability theory and related areas such as fractal
geometry. To this end, many methods have been developed since a long time in
Gaussian frames. They often rely on some underlying "nice" Hilbertian
structure, and can also require finiteness of moments of high order. Therefore,
they can hardly be transposed to frames of heavy-tailed stable probability
distributions. However, in the case of some linear non-anticipative moving
average stable fields/processes, such as the linear fractional stable sheet and
the linear multifractional stable motion, rather new wavelet strategies have
already proved to be successful in order to obtain sharp moduli of continuity
and other results on sample path behaviour. The main goal of our article is to
show that, despite the difficulties inherent in the frequency domain, such kind
of a wavelet methodology can be generalized and improved, so that it also
becomes fruitful in a general harmonizable stable setting with stationary
increments. Let us point out that there are large differences between this
harmonizable setting and the moving average stable one.
