  In this short paper we discuss how the position - scale half-space of wavelet
analysis may be cut into different regions. We discuss conditions under which
they are independent in the sense that the T\"oplitz operators associated with
their characteristic functions commute modulo smoothing operators. This shall
be used to define microlocal classes of distributions having a well defined
behavior along lines in wavelet space. This allows us the description of
singular and regular directions in distributions. As an application we discuss
elliptic regularity for these microlocal classes for domains with cusp-like
singularities.
