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In this paper, we study in-depth the problem of online self-calibration for robust and accurate visual-inertial state estimation. In particular, we first perform a complete observability analysis for visual-inertial navigation systems (VINS) with full calibration of sensing parameters, including IMU and camera intrinsics and IMU-camera spatial-temporal extrinsic calibration, along with readout time of rolling shutter (RS) cameras (if used). We investigate different inertial model variants containing IMU intrinsic parameters that encompass most commonly used models for low-cost inertial sensors. The observability analysis results prove that VINS with full sensor calibration has four unobservable directions, corresponding to the system's global yaw and translation, while all sensor calibration parameters are observable given fully-excited 6-axis motion. Moreover, we, for the first time, identify primitive degenerate motions for IMU and camera intrinsic calibration. Each degenerate motion profile will cause a set of calibration parameters to be unobservable and any combination of these degenerate motions are still degenerate. Extensive Monte-Carlo simulations and real-world experiments are performed to validate both the observability analysis and identified degenerate motions, showing that online self-calibration improves system accuracy and robustness to calibration inaccuracies. We compare the proposed online self-calibration on commonly-used IMUs against the state-of-art offline calibration toolbox Kalibr, and show that the proposed system achieves better consistency and repeatability. Based on our analysis and experimental evaluations, we also provide practical guidelines for how to perform online IMU-camera sensor self-calibration.