I would say that your question is not well posed, in that you use the term SNR for power distribution networks, but then probably you mean the distortion of the network with respect to a sinusoidal reference, i.e. Harmonic Distortion first of all, whereas I would use the term "SNR" for the measurement of voltage or current. The measurement quality, as measured by SNR of the read quantities, depends of course on the measurement system, e.g. transducers, data sampling (so number of bits of the ADC, oversampling, averaging), post processing methods ... a lot of things for one question.
Assuming that you wanted to speak of distortion of network, then as you say we may distinguish between voltage and current: a load is connected to a distribution network, this load absorbs distorted current (such as a 3-phase rectifier), that is characterized by some harmonic distortion: there are limits for individual harmonics (IHD) and for the total harmonic distortion (THD). A good reference is the IEEE Std. 519 (I attach its Table 2); as you see limits are given depending on the short circuit current ratio, that is the largest the ration, the stiffer the network, so the lower the impedance offered to the load and its current, and the lower the caused network voltage distortion.
So, we come to the voltage quality: it depends on the combined effect of all distorting loads and the amount of installed power. Usually, there are not real limits for normal loads in terms of voltage distortion (they are expressed in current, as we see above), but they are "guaranteed" a voltage with some quality, so low distortion. A standard for this is the EN 50160, but IEC 61000-2-2 is more instructive and complete (I attach its Table 1).
To conclude, when you have the amount of individual harmonics and the THD you can also calculate the SNR, because you interpret the harmonics as noise (that is not, from a signal processing standpoint, and not only that). The reciprocal of the sum of squares of the % limits gives you the SNR in power; taking the square root it will be in amplitude.
There are many references for values of network distortion or Power Quality assessment. Personally I measure Power Quality in electric transportation systems (railways, metros,...) and you can find some refs of mine.
Andrea Mariscotti Thank you for your detailed answer. I'm thinking the same way, and I can't agree more with your answer. The SNR value is used in many simulation softwares to add noise to the signal, even though it is almost never used in the distribution network analysis. However, your 4th paragraph is what I was thinking about, since this is how I planned to calculate the SNR.