Perching helps small unmanned aerial vehicles (UAVs) extend their time of operation by saving battery power. However, most strategies for UAV perching require complex maneuvering and rely on specific structures, such as rough walls for attaching or tree branches for grasping. Many strategies to perching neglect the UAV's mission such that saving battery power interrupts the mission. We suggest enabling UAVs with the capability of making and stabilizing contacts with the environment, which will allow the UAV to consume less energy while retaining its altitude, in addition to the perching capability that has been proposed before. This new capability is termed "resting." For this, we propose a modularized and actuated landing gear framework that allows stabilizing the UAV on a wide range of different structures by perching and resting. Modularization allows our framework to adapt to specific structures for resting through rapid prototyping with additive manufacturing. Actuation allows switching between different modes of perching and resting during flight and additionally enables perching by grasping. Our results show that this framework can be used to perform UAV perching and resting on a set of common structures, such as street lights and edges or corners of buildings. We show that the design is effective in reducing power consumption, promotes increased pose stability, and preserves large vision ranges while perching or resting at heights. In addition, we discuss the potential applications facilitated by our design, as well as the potential issues to be addressed for deployment in practice.