For stability impulse or step response testing is often used. A VNA (vector network analyzer, capable of showing magnitude and phase) can also show things like Nyquist plots, K-factors, and other measures of stability.
Two-tone (or multitone) testing provides more insight into both time and frequency domain performance and how/what nonlinearity terms are present. Classic distortion response in very predictable since there is a mathematical relationship between single-tone and two(multi)-tone distortion products and the order of the nonlinearity. Deviations can indicate power compression or other issues (than power compression, things like hystersis, ISI-type distortion, nonlinear frequency response, etc.) that are harder to see in a single-tone test.