I want to run 8 new circuits, four 15 amp and four 20amp. Each can be paired with a beginning run of a 14/ 3 or 12/3 wire from the breaker box to a junction box and then can be split off so that a 12/2 or 14/2 wire goes to a recepatcle, or to a device, or to lights. For example one 12/3 wire can leave the breaker box and be terminated in a junction box. Then one 12/2 wire can continue from the junction box running to an outlet for a heater. Then from the junction box, a second 12/ 2 wire can continue the second hot wire belonging to the second circuit to a bathroom GFCI. 2 breakers, two circuits, one ground and neutral wire for the first part of the run. I have seen this used for a string of kitchen counter top GFCI outlets. Is the use of a 3 wire run to lessen the amount of wire one needs to run, permitted in all general situations or are there code restrictions? For example if one part of the branch goes to a dishwasher and the other to some lights, is that okay for these sorts of things to share the ground and neutral wire at one point in the run back to the breaker box?

Thank you

Its called a Multi circuit(2 hots on opposite phases sharing a neutral).
I would use a 20/15 amp 2 pole breaker for each 12/14-3, so each that share a neutral will also be off, when 1 circuit is off. A code requirement.
If you had a black and the red of 1 cable on same phase, neutral could carry the total of both circuits.
If 1 drew 20 amps and other drew 15 amps, neutral would carry the total of the 2,(20 + 15 =35 amps).
When they are on opposite phases, the neutral draws the difference of the 2(20-15 =5 amps) neutral would carry 5 amps.

Strat... I agree with you up to a point. Since two pole breakers are internally linked, if you were to wire one room with receptacles on one pole, and lights on the other, an overloaded outlet would put you in the dark. Two pole breakers would certainly take care of the neutral being overloaded merely by design (except for some Federal Pacific panels) but careful attention to wiring will achieve the same purpose.

Best is to stay away from shared neutral circuits. Simple as that.

An broken or open neutral will cause up to 240 volts being applied to 120 volt devices, and causes irreparable damage.

They are just not worth the effort.

I understand why the two poles or breakers need to be on separate bus bars or "Phases", and how the neutral will take the difference. And I understand how if the two hot wires are not on separate bus bars or "phases" the neutral wire will take the total voltage or current. But I have not been able to see how if the mutiwire circuit is on separate bus bars/phases and then if the neutral were to be disconnected in a shared part of the run, that 240 volts could then be applied on one of the hot wires without tripping the breaker and thereby over load a device expecting to encounter only 120Vac. I think I understand how the current would drop, but not how the voltage or the amps could exceed the 20amp/120 volt each breaker or pole were suppling to the black and red hot wires respectively. My quess is I am missing some electrical theory principle involved in the way some aspect of this muti-wire (shared neutral) circuit works. Could someone say some more about it? Thank You so much!

I added a simple wiring sketch to illustrate the effect of a broken shared neutral.

You can see that two loads once in two parallel circuits are now two loads in one series circuit.

The actual volts across both loads will be dictated by the impedance (AC Resistance) of each load. If I had more time, I could show actual loads with real impedance's and the resultant voltages across each load.

This is a very real problem that does happen, more often than one would think.

I just had the problem in a bank office last week. The 200 volts at one outlet took out a security system power supply, and 160 volts took out a desktop PC.

There was no catastrophe with this problem, the devices went quietly. I have seen catastrophic damage done with devices bursting into flames from the abnormally high voltages.

I was moments from shutting down the bank and evacuating all personnel and customers, until I had a good understanding of the coverage of this problem, and that I had it under control.

jam1,

Go read this thread. https://www.askmehelpdesk.com/electrical-lighting/current-but-no-voltage-any-ideas-290529.html It may help explain current and voltage to you and then you will understand what is being said here.