Vertical lines noise in spectrogram
After wasting forever on this problem, I found that it was just a question of window function. Doh!

Window functions that are not zero-ended produce these vertical lines. For example, Rectangular Window, Hamming Window, Gaussian Window (with low sigma) produce these lines, and Barlett Window, Hanning Window, Blackmann Window, Welch Window, don't.

I had tried other window functions but maybe during my tests, I made a coding mistake so I didn't find the fix. Moral of the story : do tests carefully!