I'm still learning how to keep trees alive, mainly, and trying to soak up as much knowledge as I can. Maybe in 10 or 20 years, I'll have a knowledgeable opinion on the old rules.
Some rules are still and will always be "alive" / "important". Take a look at great Bonsai masterpieces around the world to check..
In my opinion :
1-asymmetry (same as paintings/sculptures etc) even on chokkan (vertical trunk) ;
2-depth : rear branch(es) is almost always here, sometimes not visible because of foliage or other side branches, but there.
3-proportion/balance right-left/stability : a thin slant style trunk should wear lightweight foliage/branch mass. A shakan with a small/weak nebari and big & heavy branches will look unstable/unbalanced as if it was about to fall down. Proportions between the size of the tree & the size of the leaves. Proportion between the size of the trunk & the volume of foliage (and diameter of branches) in order to make the Bonsai "credible"/well balanced.
It may vary, depending on your personnal style, but it is good to know those rules before transgressing them.
4-Rules of association between the tree and the pot. Although it has changed (and may continue to change) during time (the pots used to be bigger in old Kokufu-ten exhibitions), although it depends on countries (Taiwan Bonsai pot are sometimes more coloured/flashy), there are rules to chose a right pot and the position of the tree in the pot is based on rules too.
Dunno if we can call it a rule, but a good Bonsai always have a good trunk (tapper, character given by the bark, shari, jins, movement etc) and a good nebari/base. It is said to be the most important : trunk & base. Banches are very important too (position, size, tapper, movement)
Hope that other people here will give other "main rules" to help you get more knowledge, because those "main"/intangible rules are important ... to follow or at least to know.