In the late 90s I learned about wavelets as the "big new thing" that would replace the Fourier transform. I could be mistaken, but I don't think that has come to pass. Why not? No benefit or not as widely applicable? Less efficient to compute?

I think it was kinda like Fuzzy Logic.   Lots of initial hype, but it eventually settled into the niches where it is actually useful.

Can you elaborate on those niches?  I've seen them used in biosignal analysis (i.e. EEG), but there are likely more points of application.

The popular technical press may have billed them as replacing the Fourier Transform, but I don't think the originators pretended that to be so.

Otherwise -- what boyerkg just said.

So, I did my thesis in '95 on the Fast Wavelet Transform (FWT), and the concepts are still valid today in image compression or anywhere you need a local energy transform.  It will never replace the Fourier Transform for continuous energy signals, either cosine/sine or complex e^jwt.  I don't know if this answers your question, but the FWT is still being used.

Wavelet transforms are also integral transforms, just like the Fourier Transform. The difference is that the kernels you use are localised in time (not only in frequency). This serves two purposes:

i) you treat your signal in slices (chunks) with the advantage that for low frequencies you can take larger time intervals, without loosing resolution; and

ii) you can treat signals with discontinuities, since you are analysing them in chunks.

Wavelets are still used in many fields, they are just not as popular as the Fourier Transform.

But the same can be said for the Short-Time Fourier Transform (STFT).

But, as best as I can tell, the analysis window of a wavelet transform gets smaller for higher frequency.  In the STFT the width of the analysis window remains constant for the Fourier Transform that returns the spectrum over all frequencies of interest.

The Wavelet transform has been, and is, part of the continuing search for 'Spectral Localisation'.  As you know, The information contained in a time series f(t) and its Fourier transform F(w) is identical.  Yet some features or events vague or indistinct on the time series may be distinct, and easier to see in the frequency domain, on F(w), and vice-versa.

The Short Term Fourier Transform (STFT) is one simple and early attempt at spectral localisation.  STFT examines the time series and the spectrum in a short stretch of the time series.  However, this runs into the Uncertainty Principle, which states that you cannot localise the time information AND localise the spectral information at the same time.

STFT, Wavelet transform (WT), S-transform (ST) are just a few of the spectral localisation methods that have been developed in the last four decades.  Feature detection on 1-D data series and 2-D images is just one of the uses.  Others include data compression.

There is a certain level of technical hype of the type 'greatest thing since sliced bread' at the first discovery of a new technique.  With maturity the hype dies down.  So Wavelet transform is alive and well, but without the hype.  Possibly you do not hear of spectral localisation, Wavelet transform or the S-transform, because you have no specific need for these methods.  In other disciplines with the need for spectral localisation, these techniques are being used and developed.

Possibly you do not hear of spectral localisation, Wavelet transform or the S-transform, because you have no specific need for these methods.

You are too kind. Perhaps I have no need, but I do not understand them well enough to say that for sure. Thank you for mentioning the S-transform. Wikipedia seems to indicate that it is the best of both Fourier and Wavelet Transforms. Am I understanding that correctly?

Here is a quote from Brown, Lauzon, and Frayne's IEEE paper from 2010, which also makes the S-transform sound superior, at least to a layman like me:

While the original Fourier transform (FT) is an extremely important signal and image analysis tool, it assumes that a signal is stationary, i.e., that the frequency content is constant at all times in a signal, or at all locations in an image. Since most interesting signals are nonstationary, a series of techniques have been developed to characterize signals with dynamic frequency content. These methodologies are the foundation of the field of time-frequency analysis.

A simple approach to the problem of nonstationary signal analysis is the short-time Fourier transform (STFT). In this technique changes in frequency over time are captured by using a window function to provide time localization. However, the choice of window function represents a compromise. Narrower windows provide better time resolution but poorer frequency resolution, while wider windows provide the converse. Ideally, the window width is chosen to produce the best representation of particular features of interest in the signal, but this requires a priori knowledge.

The wavelet transform (WT), which has been applied to a wide variety of signal processing problems, improves on the STFT by introducing the concept of progressive resolution. The WT provides the equivalent of finer time resolution at high frequencies and finer frequency resolution at low frequencies. However, the WT does not measure frequency but only an analogue, called scale. Additionally, the WT provides either no phase information, or phase measurements which are all relative to different local references. This is in contrast to the conventional concept of phase, as provided by the FT, where all phase measurements are relative to a global reference.

The S-transform (ST) exhibits globally referenced phase and frequency measurements similar to those of the DFT and STFT, as well as the progressive resolution of the WT. This combination of desirable features has shown promise in a wide variety of applications, including detecting abnormalities in the heart, identifying genetic abnormalities in brain tumors, analyzing electroencephalograms, transmitting medical images, characterizing the behaviour of liquid crystals, detecting disturbances in electrical power distribution networks, monitoring high altitude wind patterns and detecting gravitational waves.

One discipline that has taken to Wavelets, with significant contributions, is mathematics.  See the Wikipedia entry for Baroness Ingrid Daubechies, a mathematician from Belgium. https://en.wikipedia.org/wiki/Ingrid_Daubechies .

My own inclination is towards S-transform, not Wavelets.  Regardless, research into all manners of time-frequency (or spectral) localisation methods are alive and kicking.