Wow! What country & area are you in?
It would be interesting to know.
Is this probably even a three phase system?
Are they 120 degrees apart?
Could each wire be just one of three independent single phase systems since the power company has no control over people unbalancing the loads?
You can easily [tm] find out.
See section "Method" below.
3 phase power when 1 or 2 phases is missing is not very good for 3 phase equipment so that does not sound like a very useful choice.
You didn't ask, but you could easily make an "auto-swapper" using relays that would deliver power to the output from one of whatever phases were available. Don't tell your neighbours - if everyone had these they would probably dance together as the power company struggled to balance the system :-).
An auto-swapper could be built with one relay with 3 changeover contacts plus one relay with one changeover contact.
Vout = A + B/A + C/A/B
If A is alive, use it via an A operated contact.
If A is dead the A operated contact blocks A.
If A is dead use B if B is alive. (B contact and /A contact)
If A is dead and B is dead use C. (C contact + /B contact.)
If /A and /B and /C, go for a walk.
C could also have a C operated contact if you don't want C connected when dead.
Some equipment may not like power being disconnected and connected again at relay-operate speeds.
METHOD:
WARNING
A meter used for the following test MUST be rated for 400V mains if you are on a 230 VAC or similar system. NOT JUST "has a 400V capable range" .
Usually people will say "CAT 3" or "CAT 4" but even that is misleading.
This paper The myths of instruments and safety addresses up a few comminly held and potentially dangerous ideas.
For now, understand that the measurement I am about to suggest involes 400 VAC in a 230 VAC system and 200 VAC in a 110 VAC system. Your meter and leads and anything electrically connected must withstand mains PLUS any spikes that may be present. Given the situation you describe, spikes and worse could be quite likely. Be aware that arc over in an under-rated multimeter can kill you and people have died in such circumstances, even measuring 230 VAC!.
You could also use a resistive divider. They too can have issues.
If proceeding ... :-)
Name you phases A B C.
Measure phase to phase to phase in all combinations. AB BC CA.
If two readings are about 1.7 x Vmains (200V or 400V for 110VAC/230VAC systems) you have two phases of a 3 phase system.
If 3 readings are ~= 1.7 x Vmains you have 3 phases.
If readings are ~= 0V or ~= 2 x Vmains (220 VAC or 460 VAC you have independent single phases from separate transformers.
If they are in phase you get about 0V. This would be usual.
If you get about 2 x maians they are out of phase. This would be unusual due to construction methods.
I said "independant separate phases" meaning that they may have two transformers with 3 phases R G Y phases. If you have R from transformer 1 on your phase A, and R from transformer 2 on your phase B then you will probably get 0V between them.
They COULD even feed you R G Y phases from 3 seperate transformers. You could use that as 3 phase power but the current flows would be "interesting".
1) Why would you expect a potential difference not to appear? N and earth are tied together only in distribution/transformer boxes, that's a long way till your house, the N wire is usually carrying some current so its potential might slightly differ from ground.
2) If you call tripping the whole house an issue, well that's an issue then. I'd say that safety is not compromised at all, it is instead enforced by this behaviour. Disconnecting also the N through the circuit breaker would work.
Best Answer
Perhaps the below sketch will help. This is earthed according to Australian practice (AS 3000).
Note that the dirt ('general mass of the earth') is high impedance and no appreciable current flows between the earth electrodes. Therefore, any current flowing via the active conductor must return via the neutral conductor, since there is no other possible path for it to flow.
Q: "If there is no appreciable current between the earth electrodes, then why is it dangerous to touch a live wire?"
A: Touching a live wire does not, itself, kill you. Note that birds are quite happy to sit on powerlines.
What kills you is completing a circuit between active and earth (or neutral). A very direct way of doing this is shown below. Note that many things are bonded to earth - the steel frame of your house, the reinforcement bars in concrete slabs, plumbing (metallic water services must be bonded to earth in Australia), and so on.
You don't even have to touch two metallic conductors to be electrocuted. Ground has 'high' resistance but not infinite resistance. Therefore, when a live wire touches the ground, a current diffuses into the ground, eventually returning to the source. If you present a lower resistance path for this current, it will flow through you. This is the concept of 'touch voltage' and 'step voltage' which we must consider when designing high-voltage installations.
Touch voltage is very bad, because if you touch a live conductor with your hands, and the current flows to your feet, it will traverse across your chest i.e. your heart. The heart is very sensitive to electric shock.
Step voltage is not so bad, because the current flows into one leg and out the other. There are no really important organs between your legs.
As an aside - we have had incidents where crane drivers have contacted HV overhead lines. The recommended action is to stay in the crane as the surrounding metal is all at the same voltage, so you won't be electrocuted. If you do have to run away, i.e. because the crane is on fire, the recommended action is to jump off (so you don't touch the metal of the crane, and the ground at the same time - they may be at different voltages) and then bunny hop away with your feet close together. Keeping your feet close together minimises the voltage between your feet.
Yes, I am serious.