What are the advantages of using a 120v/240v split phase electrical system vs a 127v/220v 3 phase electrical system? Why did some 60hz countries choose to use 120v/240v split phase electrical systems while others chose to use 127v/220v  3 phase electrical systems?

For small consumers like family houses or so, who uses their "own" transformer, the split phase suffices with just a simple, single coil cheaper single phase transformer. The 3-phase transformers are way more complex (at least 3 pillars, 70% more winding for the same power if they are supposed to tolerate unbalanced loads).

The 3 phases are then more economic when there is one common transformer (HV->3x127/220) feeding many customers, where the imbalance somewhat averages out and where the total load is so high it may impact the upstream distribution when only at 1 phase. So more of a larger building installation. Plus higher power machinery (HVAC, elevators, water management,...) of larger buildings benefits the most (in the form of simplicity, reliability and efficiency) when directly using the 3 phase power.

Here in Israel, we use 240V/380V 3 phase systems, that comes from a single 3 phase transformer.
You can see such a 3 phase transformer here: https://www.lighting-gallery.net/gallery/displayimage.php?pos=-187919

Here in the UK we have a 3-phase distribution system, which normally gets fed to consumers at 415/240V (415V between phases, 240V between any one phase and neutral). This is officially 400/230V, but in practice it is still the older voltages that are found, as it was purely a paper excercise to "harmonise" voltages across Europe.

Large customers get all three phases, so they can balance their loads across them, but most domestic customers only get one of the three phases, usually on an incoming fuse of somewhere between 60 and 100A. The three phases are balanced out from the substation transformer by putting each house on the next available phase, so house one will be on phase A, house 2 on phase B and house 3 on phase C, and so on.

We'd love a 3-phase supply, I suppose technically we could have one, if we were prepared to pay for it, but it wouldn't be cheap and as we rent I don't suppose the council would be too keen on it either 

For residential, present day the maximum allowed single phase load is 16A, larger devices require to use 3 phase.
The only exception are apartment units, with 1x25A in a single phase. And that is pretty much the maximum power service you may get at single phase, any higher power rated service is only 3 phase.
The reason is very strict enforcing of phase-phase load balance here, probably to avoid idling losses from oversized distribution transformers (transformer optimized for low losses at symmetrical load uses to exhibit poor tolerance towards load asymmetry; transformers designed for asymmetry tolerance exhibit higher idle, as well as high load losses even when loaded symmetrically, so distribution companies do not like to install them).

So a single phase you may get for a small apartment, a garage cell (here garages are often separate buildings) or some really low power equipment (like some telecom node,...), the appartment may get 1x25A, the other single phase service just 1x16A max.
Larger apartment, mainly when the house was build without gas for stoves or used electric water heaters, family homes, all others are strictly 3 phase.
And even with rather small appartments build in 60's, the electrical overhauls more and more often means a switch to 3 phase (3x25A or so). And even when the customer does not want to switch yet, the consumer cable (from the meter shaft to the breaker panel) installed is already the 5x4mm^2 (so ready for the 3x25A).

I don't think they really want to put domestic consumers on 3-phase supplies here in case some idiot gets the phases mixed up and zaps themselves with 400V+. I don't know, I'm speculating, but that's how it's always been done here. Each house along the street is fed from a different phase, so it tends to balance out. We're in a block of 6 flats, so there is a 3-phase feed into the building, but it's still a single phase to each one, so there are 2 flats on each phase in the block.

The largest single load we have is the shower, at 8kW, so allow 35A, it has a 40A breaker on the main panel. The next is the cooker, which is rated for a 32A supply feed, but of course it's never going to take anything like that unless you switch all the stove-top burners on AND both ovens all at once, which we're not likely to ever do.

The power sockets are on two 32A ring circuits, one for the kitchen and one for the rest of the flat. Most of the loads on that are computers and the lamps that we run up, so not a lot most of the time, a few hundred watts if that, maybe 1 or 2kW if we have a couple of high power lamps lit as well. Or the kettle, of course, which is 3kW. I'm not a great fan of the ring circuit and most UK electricians won't fit them any more, preferring 20A radials, but there are still many tens of thousands of them in use up and down the country.

All that's left are 2 6A lighting radials, which don't take a lot at all.

I think our main fuse in the supply panel outside is 80A, but I'm not sure, I know the main breaker is rated 100A.

I wanted to question the question with: Who uses 220Y/127V?
I never heard of it!  I believed all of North America (Canada, Mexico, USA) have 208Y/120V service.
So I just looked into it, and I am gobsmacked.  Mexico has 220Y/127V.  The three North American countries are not all alike!

The root-of-three proportions 127:220:380 is really neat.  I cannot believe I never read about it before.  Thank you for making me aware of this, WorldwideHIDCollectorUSA.

I am aware of 127v/220v power systems being used in Mexico, Brazil, and Saudi Arabia as far as I can tell. In addition, all of the Isthmus countries and some of the northernmost South American countries like Venezuela and Colombia also seem to use 120v/208v and 120v/240v power as well. Interestingly, in Brazil, some of the appliances can use either 127v or 220v mains and are known as “Bivolt” devices over there.

 When  I visit nuevo Larado boarder with Texas.  I have seen somthing A bit different  with one single phase pole transformer powering  A city block.  The home have two edison base 30 amp fuses for Mains!   Highest rating fuse in sub panel is 15 for swamp
Cooler and bunch Of 6 amp edison base fuse for lights and fans. 
  They it was 120/240 volt single phase.!