Are there any advantages oversampling?

Let me suppose that by Nyquist, you mean that there is some frequency $F_s$ such that there is a existence theorem that says:

if one samples any continuous $s(t)$ at $F_s$ (or above), $F_s$ denoting an extremal frequency or a extremal bandwidth, to get a discrete signal $s[k]$, then one can recover $s(t)$ by some algorithm.

• Case 0: neither oversampling nor undersampling, you sample at $F=F_s$. Theoretically you are safe, but sampling systems are never perfect (noise, quantization, jitter, lag, limited time), and reconstruction algorithms not often stable, so there is a risk that some of the signals you sample, you lose information (like with aliasing)
• Case 1: oversampling , you sample at $F>F_s$. Theoretically you are safer, and many systems specify that 10% or 20% above $F_s$ are safe bets for relatively clean signals. Higher, you can hope to have more chance to retrieve weak signals in noise, etc. but the higher the rate, the larger the signal to store and manipulate, and, sometimes, (potentially ) better is worse.
• Case 2: undersampling , you sample at $F<F_s$. Theoretically you will lose information. But if the signals you are interested in are not "all potential signals" satisfying the limits, but a restricted set, or if the processing you do is robust, or the features you extract are not so demanding, then with high probably, you will get what you want at a cheaper price.

1. Nyqusit Frequency simply means "half the sample rate"
2. Undersampling means your signal bandwidth is higher than the Nyquist Frequency. You get aliasing
3. Oversampling means your signal bandwidth is lower than the Nyquist Frequency. You don't get aliasing
4. Most practical applications choose a Nyquist Frequency that's slightly higher than the highest frequency of interest. That gives some space to manage the transition band between the signal frequencies and the Nyqusit Frequency (at which the signal should be 0). For example audio is considered to have 20 kHz bandwidth but the typical Nyquist frequencies are 22.05 kHz or 24 kHz