What can we do to analyze very noisy dynamical data, when spectrum seems ok to human?

Are there any relative simple methods to do that?
I have few signals to analyze. Using audacity sound program, in spectrum view i can visually recognize signal frequencies change over time, or hear it when playing the sound. But when try to do any DSP algorithm, it looks impossible. What is the secret of human perception, that makes it possible to hear signals below noise level?
I tried FFT with different windowing and size, tried to perform Hough transform on the spectrum to find signal with approximate formula. And there is no satisfactory results. For hough transform it is impossible to appropriately binarize the spectrogram. FFT maximums most time are noise maximums, and not signal. The only a little better than nothing result is convolution with approximate formulas, but it is too slow approach, because signal must be convoluted with a full range of function parameters, it can take hours to analyse sweeping arguments and do multiplications.

The "below noise level" point suggests that you aren't extracting the right signal parameters.

Are you analyzing artificial signals with known characteristic or natural signals?