The single-fiber composite (SFC) has been widely used to quantify fiber strength and fiber-matrix interfacial properties of fiber-reinforced composites. Here, a numerical model with an embedded-process-zone model to permit both interface debonding and matrix cracking is used to predict the fragmentation process and the microscopic damage around fiber breaks in SFC tests as a function of the interface strength and toughness. For low interface strengths, interface debonding occurs. For intermediate interface strengths, matrix cracks occur and delay debonding. For high interface strengths, debonding does not occur and deformation is controlled by a matrix shear, with strain hardening playing an important role. Interface toughness plays a secondary role in determining the transitions in damage modes. Well-established models assuming a constant interfacial shear strength can fit SFC data for low interface strengths, but the interface strength parameter is unrelated to the actual shear strength. In the high-strength regime, a strain-hardening shear-lag model can fit the SFC data quite well. Overall, the fiber strength distribution can be obtained from SFC tests by fitting to the fragment length versus applied strain, but estimation of interfacial properties is difficult due to the transition in dominant deformation and damage mechanisms, including matrix cracking.