Spectral estimation methods typically assume stationarity and uniform spacing between samples of data. The non-stationarity of real data is usually accommodated by windowing methods, while the lack of uniformly-spaced samples is typically addressed by methods that ``fill in'' the data in some way. This paper presents a new approach to both of these problems: we use a Bayesian framework, which includes a non-stationary Kalman filter, to jointly estimate all spectral coefficients instantaneously. The new method works regardless whether the samples are evenly or unevenly spaced; moreover, it provides a new approach to enabling processing when it is desirable to virtually eliminate aliasing by unevenly sampling. An amplitude-preservation property of the new method can be used to detect if aliasing occurred. Finally, we propose an efficient algorithm for sparsifying the spectrum estimates when we know a priori that the signal is narrow-band in the frequency domain. We illustrate the new method on several data sets, showing that it can perform well on unevenly sampled nonstationary signals without the use of any sliding window, that it can estimate frequency components beyond half of the average sampling frequency when the signal is unevenly sampled, and that it can provide more accurate estimation than several other important recent and classical methods.