  We quantify the instability towards the formation of multipolar states in
coupled spin-1/2 chain systems with a frustrating J1-J2 exchange, in parameter
regimes that are of directly relevance to edge-shared cuprate spin-chain
compounds. Three representative types of inter-chain coupling and the presence
of uniaxial exchange anisotropy are considered. The magnetic phase diagrams are
determined by Density Matrix Renormalization Group calculations and completed
by exact analytic results for the nematic and dipolar phases. We establish that
the residual couplings strongly affect the pitch of spiral states and their
instability to multipolar phases. Our theoretical results bring to the fore
novel candidate materials close to quantum nematic/triatic ordering.
