I am most familiar with time series forecasting methods, so my examples will reflect this.
Many, if not most classical forecasting methods will give you a predictive distribution, usually based on a Gaussian assumption, often simply assuming homoskedasticity and estimating the variance in a very naive way. This is where classical ARIMA and exponential smoothing methods get their prediction intervals from. Most forecasting textbooks unfortunately only hint at this.
(G)ARCH is explicitly designed to focus on dynamics in the second moment. It is usually again paired with a normal distribution in estimation, and thus could also be used to derive predictive densities. This is - again implicitly - usually used in Value at Risk forecasting. Nassim Nicholas Taleb essentially claims to have gotten rich off using better distributional assumptions than the normal.
One neural network architecture that explicitly aims at full predictive densities is DeepAR (Salinas et al., 2020). The authors work for Amazon; retailers have a very strong interest in density (or at least quantile) forecasting to set safety amounts. A few years back, Feindt et al. published a little about NeuroBayes, which used neural networks for density forecasting in a Bayesian paradigm; however, it looks like ever since he founded BlueYonder, development has not been published.
Apart from the time series context, of course the simplest example might be OLS and the predictive distributions given by conditional means, (homoskedastic) variance estimates and a t distribution.
A very nice introduction to probabilistic prediction, mainly from the point of view of evaluating such predictions, is Gneiting & Katzfuss (2014).