  The geometrical critical behaviour of the two-dimensional Q-state Potts model
is usually studied in terms of the Fortuin-Kasteleyn (FK) clusters, or their
surrounding loops. In this paper we study a quite different geometrical object:
the spin clusters, defined as connected domains where the spin takes a constant
value. Unlike the usual loops, the domain walls separating different spin
clusters can cross and branch. Moreover, they come in two versions, "thin" or
"thick", depending on whether they separate spin clusters of different or
identical colours. For these reasons their critical behaviour is different
from, and richer than, those of FK clusters. We develop a transfer matrix
technique enabling the formulation and numerical study of spin clusters even
when Q is not an integer. We further identify geometrically the crossing events
which give rise to conformal correlation functions. We study the critical
behaviour both in the bulk, and at a boundary with free, fixed, or mixed
boundary conditions. This leads to infinite series of fundamental critical
exponents, h_{l_1-l_2,2 l_1} in the bulk and h_{1+2(l_1-l_2),1+4 l_1} at the
boundary, valid for 0 <= Q <= 4, that describe the insertion of l_1 thin and
l_2 thick domain walls. We argue that these exponents imply that the domain
walls are `thin' and `thick' also in the continuum limit. A special case of the
bulk exponents is derived analytically from a massless scattering approach.
