This paper is concerned with the study of linear geometric rigidity of shallow thin domains under zero Dirichlet boundary conditions on the displacement field on the thin edge of the domain. A ribbon is a thin domain that has in-plane dimensions of order O(1) and epsilon, where epsilon is an element of(h,1) is a parameter (here h is the thickness of the domains). The problem has been solved by Grabovsky and the second author in (Ann de l'Inst Henri Poincare (C) An Non Lin 2018 35(1):267-282, 2018) and by the second author in (Arch Ration Mech Anal 226(2):743-766, 2017) for the case epsilon=1, with the outcome of the optimal constant C similar to h-3/2 ,C similar to h-4/3, and C similar to h(-1) for parabolic, hyperbolic, and elliptic thin domains respectively. We prove in the present work that in fact there are two distinctive scaling regimes epsilon is an element of(h,h] and epsilon is an element of(h,1), such that in each of which the thin domain rigidity is given by a certain formula in h and epsilon. An interesting new phenomenon is that in the first (small parameter) regime epsilon is an element of(h, root h], the rigidity does not depend on the curvature of the thin domain mid-surface.