Fragments can have any manner of variable affecting the worlds within, which in turn affect game boards. The three Rules however generally remain consistent and will always manifest unless they are overcome or through other interference. In many cases, one Rule manifests because of the effects of another Rule.
The Rules of Higurashi
This article or section contains untagged major spoilers for all of the original Higurashi no Naku Koro ni, possibly including the console-exclusive arcs. Readers who have not completed the main story are advised not to proceed further.
The three Rules are as follows:
A random person becomes violent under the influence of paranoid thoughts.
At least one person, usually Keiichi Maebara, Shion Sonozaki. or Rena Ryuugu, will succumb to paranoia and eventually kill some people. This can be caused by Rule Y, Rule Z, or will manifest interdependently.
Tomitake is always found dead in June of 1983, and Takano will follow suit either by disappearing or being found dead herself. As they are the fifth victims of the curse, suspicion falls onto the Sonozaki family, long suspected of masterminding it every year.
The Sonozaki family always acts as if they are responsible, no matter if they are or not.
The Sonozaki family follows a precept that anything that happens in the village, the Sonozakis must act as though they were behind it all along, making it seem as though they have lots of influence. The result is that many people, including Kuraudo Ooishi, believe they are responsible for Oyashiro-sama's curse and will spread this idea to other people, sometimes resulting in Rule X taking effect.
The Rules of Umineko
Umineko no Naku Koro ni's Rules are explored in the side story Bernkastel's Letter and revealed explicitly in the Twilight of the Golden Witch manga. The manga also introduces the idea that Rules remain consistent so long as the Game Master keeps them consistent, as revealed by Battler's explanation of the fifth game, which had a different Game Master than the previous one.
See the Writing Process section of Message Bottles for specific information.