The properties of freestanding tensionless interfaces and membranes at low bending rigidity kappa are dominated by strong fluctuations and self-avoidance and are thus outside the range of standard perturbative analysis. We analyze this regime by a simple discretized, self-avoiding membrane model on a frame subject to periodic boundary conditions by use of Monte Carlo simulation and dynamically triangulated surface techniques. We find that at low bending rigidities, the membrane properties fall into three regimes: Below the collapse transition kappa(BP) it is subject to branched polymer instability where the framed surface is not defined, in a range below a threshold rigidity kappa(c) the conformational correlation function are characterized by power-law behavior with a continuously varying exponent alpha, 2 < alpha