Semiconductor nanomembranes (NMs) have emerged as an attractive nanomaterial for advanced electronic and photonic devices with attractive features such as transferability and flexibility, enabling heterogeneous integration of multi-functional components. Here, we demonstrate transferable single-layer GeSn NM resonant-cavity-enhanced photodetectors for 2 μm optical communication and multi-spectral short-wave infrared sensing/imaging applications. The single-layer strain-free GeSn NMs with an Sn concentration of 10% are released from a high-quality GeSn-on-insulator (GSOI) substrate with the defective interface regions removed. By transferring the GeSn NMs onto a predesigned distribution Bragg reflector (DBR)/Si substrate, a vertical microcavity is integrated into the device to enhance the light-matter interaction in the GeSn NM. With the integrated cavity and high-quality single-layer GeSn NM, a record responsivity of 0.51 A W-1 at 2 μm wavelength at room temperature is obtained, which is more than two orders of magnitude higher than the reported values of the multiple-layer GeSn membrane photodetectors without cavities. The potential of the device for multi-spectral photodetection is demonstrated by tuning the responsivity spectrum with different NM thicknesses. Theoretical simulations are utilized to analyze and verify the mechanisms of responsivity enhancement. The approach can be applied to other GeSn-NM-based active devices, such as electro-absorption modulators or light emitters, presenting a new pathway towards heterogeneous group-IV photonic integrated circuits with miniaturized devices.