It depends a lot on which product needs to be examined.
for Benzene the freezing point is a first line test to know something about the purity,
For some amines it may be the boiling-point ( as seen in GC analysis)
and a nice example of a melting-point as a quality indicator ( as was used in the 80ies ) is that of ε-Caprolactam . which is the basis to make nylon 6,6
In a test tube a few gram of Capro was put in a heated in a heating block till +- 72°C; with a very precise mercury thermometer in it. When it was completely melted we took it in our hands and let it cool down, observing the temp, and let it go down below it’s point of solidification/melting-point, but it stayed liquid = this was the phenomena of ‘under-cooling’.
Then with the thermometer we made a little scratch on the inside and all of a sudden it started to turn solid, forming kind of crystals, what also happened on the same time was the temp from like +- 68,5°c went up to that actual melting- point, and we read that and noted in the books to 2 numbers behind the decimal point !, That few hundreds of a °C made the difference. 69.10 and above was good to very good, for most purposes 69.08 was acceptable but like 69.05 and lower was not high enough, and was immediately reported to the production leader.