Looking for a case hardness implies a carburizing or other surface treatment of steel (possibly nitriding), where the hardness differs based on the distance from the surface. Case depth is going to be measured in thousandths of an inch in the US. The hardness scales to use for hardness in the case would be either Vickers or Knoop - both microscopic scales. Even the superficial Rockwell scales would measure on too macroscopic a scale to accurately measure hardness in the case. Depending on the base steel, Rockwell B or Rockwell C could be used to measure the hardness of the core material. A typical grade for case hardening in the US would be 8620.
Typical specifications I've seen in the past for carburized steels specify a minimum hardness at a given case depth, and a minimum hardness for the core material as well. Sometimes there are microstructure requirements as well, or minimum surface carbon contents.
Once you have measured a Vickers or Knoop hardness, there are conversion tables available to "convert" the microhardness to an equivalent macroscopic hardness values such as Rockwell C, Rockwell B, Brinell, etc. with caveats as to the accuracy of the conversion.
One uses different hardness measurements depending on customer and specification requirements, the nature of the material being tested - iron, copper, aluminum, hardened/unhardened, macroscopic vs. microscopic requirements, thickness of material being tested, etc.
There are 6 Rockwell superficial hardness scales (15, 30, & 45 N and 15, 30, & 45 T) about 9 macroscopic Rockwell scales (C, B, A, D, E, F, G, H, & K) a different penetrator and different load for each one - 1/16 " tungsten carbide ball, 1/8" tungsten carbide ball, & diamond pyramid indenter with loads of 60, 100, & 150 kg. I don't do the microscopic ones often enough to remember, and Brineel hardness parallels macroscopic Rockwell - different indenters, different loads, etc. There are also unique hardness measurements - Scleroscope comes to mind - rebounded ball method used mostly to measure surface hardness of rolls such as those used to flat roll steel.
Hope the above ramble helps some.