I published a blog post about Quantile Regression application to finding local extrema -- see Finding local extrema in noisy data using Quantile Regression .
The algorithm has the following steps:
Fit a polynomial through the data (using QuantileRegression).
Find the local extrema of the fitted polynomial. (We will call them fit estimated extrema.)
Around each of the fit estimated extrema find the most extreme point in the data by nearest neighbors search (by using Nearest).
Here is an example that shows noisy oscillatory data, the fitted regression quantiles, and the estimated local extrema:
Here is another example with more timid data: