daf bit: Beitzah 11

Beit Shammai and Beit Hillel are debating forbidden labors on festivals. Beit Shammai say that you cannot take a pestle¹ to use to cut meat on it, while Beit Hillel permits this. Cutting meat is permitted; they're just arguing about the pestle. What is the issue here? The pestle's normal use is for pounding grain, which is an activity that is forbidden on festivals, and if an activity is forbidden, so are its usual tools (e.g. if writing is forbidden you don't even touch pens). Here we have a case of a permitted use involving a forbidden instrument; Shammai is strict and says no while Hillel is lenient and says yes. Another dispute: you can flay a hide on a festival but not tan it; Shammai does not allow one to place a freshly-flayed hide as a doormat because treading on it contributes to tanning it, while Hillel permits. (11a)

¹ An instrument designed for grinding grain; they're talking about larger ones than what's in your kitchen today, I think, so it could actually be feasible to use it as a cutting surface.

(I don't know if hides were commonly used as doormats at this time or if this is a what-if case.)

The disputes between Beit Shammai and Beit Hillel are probably the most famous in the talmud. Shammai is almost always stricter. The halacha almost always follows Hillel. But, that said, the talmud isn't the last word on halacha, so be careful drawing conclusions about what you personally can do on festivals. Or just use a cutting-board. :-)