The properties of freestanding tensionless interfaces and membranes at low bending rigidity κ 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 κ_{BP} it is subject to branched polymer instability where the framed surface is not defined, in a range below a threshold rigidity κ_{c} the conformational correlation function are characterized by power-law behavior with a continuously varying exponent α, 2<α≤4 and above κ_{c}, α=4 characteristic for linearized bending excitations. Response functions specific heat and area compressibility display pronounced peaks close to κ_{c}. The results may be important for the description of soft interface systems, such as microemulsions and membranes with in-plane cooperative phenomena.