Estimating number of sinusoid from spectral estimation
The best approach and the details of how to do it depend greatly on the particulars of your situation.  If all your vectors are of the same length, and all your sinusoids have an integer number of cycles per vector (and therefore their frequencies correspond to a DFT bin) you can take the DFT of each vector.  Add the vectors coherently if the phases are the same in each random realization, add the DFT magnitudes if the phases are not the same.  Then set a threshold and find the points above the threshold.

If the frequencies are not on DFT bins but are spaced apart from each other, you need to modify the above approach to take account of the fact that the energy of each sinusoid is spread over multiple DFT bins.  You can still use threshold-based detection but will need to interpolate to find the frequencies.

If the frequencies could be spaced very closely together - which, from your plot, appears to be the case - the DFT-based approach is difficult.  The best approach may be a subspace-based technique like ESPRIT (Estimation of Signal Parameters Via Rotational Invariance Technique).  This involves estimating the covariance matrix and computing its eigendecomposition, then using the eigenvalues to estimate the power of each sinusoid and the eigenvectors to estimate the frequencies.  See, for examples: Richard Roy, Thomas Kailath: "Estimation of Signal Parameters via Rotational Invariance Techniques." IEEE Transactions on Acoustics, Speech, and Signal Processing.  Vol 37, No 7, July 1989.