Structural Health Monitoring (SHM), is a very active research field, aimed at producing methodologies for the periodic, and often online, assessment of structures. While there are different approaches, and perhaps definitions for SHM, there is one common necessary element if applied in practice: data acquired from sensors. In an ideal scenario, sensors could be deployed anywhere on a structure of interest, and could be added retrospectively, if not included in the original design. In practice, there are limitations to the availability and installation of sensors, such as physical access and cost. Virtual sensing has thus been proposed as a solution to the problem of sensor availability and different approaches have been applied in various fields. This paper explores and compares two approaches for virtual sensing in structural dynamics: Kalman filters and Gaussian process regression - with the ultimate purpose of assessing their suitability for SHM. The approaches are demonstrated on data from a laboratory structure: a nonlinear Brake-Reuß beam.