In the academic literature it is extremely widely applied in the last 20 years. I would estimate maybe 200 empirical papers, or more. For example a common finding is that higher frequency (daily) wavelet correlations have been high since 2007, attributable either to increasing financial interation or the financial crisis. It is also popular to estimate the time-varying variance as a non-parametric alternative to GARCH models. There are costs and benefits to this, sometimes we want to impose a hypothesis of a functional form onto the data if we have pre-existing knowledge we want to incorporate to improve generalisation of our estimate, or to ensure we are not fitting noise. You can also combine the two approaches; wavelets for the pass band filter and a parametric model for the volatility estimates. This will bais your test though.
Specifically wavelet analysis and not Fourier analysis is popular due to the non-stationarity of the mean and variance in the time domain and other characeristics that vary in the frequency domain over time. Fitting a linear combination of trig basis functions is just not going to give you good results in finance. Autocorrelation also varies in the frequency domain. It is also prefered to STFT so that we do not have to assume stationarity within a time-band. Wavelets are intuitively nice with typical process assumptions in stochastic finance that assume a fractal like property of the price process (stemming from the assumption that our price is some function of a Wiener process). This is incorrect in practice due to phenomena such as the Epps effect and the discreteness of markets imposed by latency and microstructure, but for the time scales of interest to empirical researchers it is acceptable. Indeed these ultra high frequency crystals are generally not estimated for both computation and redundancy reasons (yet there are some papers; the unbiasedness of the white noise hypothesis test I am unsure of). Another reason is that Datastream and Bloomberg are the go-to databases and they are shocking for high frequency data.
Other findings surround frequency-dependent Granger causality testing, changepoint analysis and other analyses.
It is also commonly used to analyse lead-lag relationships through the phase difference in the Morlet wavelet; since the complex sinusoid is fit you can compare the real and imaginary parts of the cofficient through the inverse tangent function to get a phase difference. This isn't a causal wavelet though so you will not be able to construct a time-frequency localised pair trading strategy with this unless your strategy can tolerate the edge effects, which I find to be highly dubious. Also, this is just for the bivariate setting.
All in all, good if your goal is to get a qualitative understanding of interrelationships and correlations in the past, not necessarily good if you are trying to extrapolate into the future since it is not built to be able to generalise. However, you can use it as part of something else if you want to extrapolate, but I am not allowed to share this.