If you upgrade both segments on a connecting flight with PlusPoints, United will only charge PlusPoints once, at the same rate as the most expensive segment.

But ... what exactly is a connecting flight?

I recently flew the following route and had to pay PlusPoints twice (=2x20PP):

Tuesday
JAC DEN 6:09pm-7:39pm
Wednesday
DEN LGA 5:35pm-11:16pm

Why did I have to? Or asked another way, how would I have had to fly so that I only had to pay 20PP once?

A flight with one or more transfers, but no stopover.

You had a stopover at DEN. To be considered a connection, you would have needed to board a flight to your destination within 4 hours of arrival at DEN. You might have been able to squeak by with a longer transfer as long as it wasn't overnight. (The PlusPoints / luggage check-through calculation is less strict than the fare calculation).

If I�m flying from London to Sydney with a transfer in San Francisco, does that use one or two sets of PlusPoints. In other words, if I buy a ticket LHR to SYD in Premium Plus, can I get the whole routing in Polaris for 30 PlusPoints?

Yes, assuming the transfer isn't overnight. If you're arriving in SFO in the morning and departing later that morning / afternoon, then one award will cover the entire distance.

I tried to see how much it would be to upgrade YVR-DEN-ORD-DTW (no long layovers), and United said it would be 20 PlusPoints per segment. There may have been consideration for connecting flights before, but it appears that no longer exists.

A mileage upgrade would still work as a single award.

I think you're reading the screen wrong. If you select each leg, it should still only total 20. You'll be charged the highest amount of any of the legs that clear. In your case, if you select all three, you'd be charged 20 if any of them cleared -- even if it was just ORD-DTW. There's even a big note to that effect on the selection page:

It should say 20 next to each segment, but if you select them all it should say 20 for the total as well.

edit: too slow! bogged down by the cloudflare!

Well ... it seems to depend on how you book the ticket?

Sort of. If you book the multi-city version and then apply +P after, IME it will usually wrap it together as one request. In that scenario, though, I'm not sure why you'd do multi-city when you can find the fare without.

Search JAC to LGA as regular (don't get cute with multi-city)
Look for the flight via DEN (or another connecting gateway under 4 hr between scheduled arrival and scheduled departure)
Review and book flight
Apply 20 Plus Points from JAC to LGA
Even it the short segment, JAC to DEN, for example, clears, and DEN to LGA does not clear, 20 Plus Points would be deducted. If one or the other clears, 20 points; if both clear, still 20 points.

I could be totally wrong here but I think you get charged the extra PP if you have a fare break, I.e. if the two segments are priced as two different one-way fares (which you can sometimes get on a multi-city search) versus a �through fare� with a connection.

I recently had a flight from a city (let�s call it XXX) served by United Express to ORD to AMS with the XXX-ORD leg on an (economy-only) CRJ and the ORD-AMS upgraded with pluspoints.

When I called to move my XXX-ORD leg to another flight that had a first class cabin, I was told I would need to apply additional PlusPoints to get that segment in First (I class was actually available) because it was priced as two separate fares XXX-ORD and ORD-AMS.

I didn�t bother and the upgrade ended up clearing anyway.

Again that�s just one data point.

You are.
You need PZ, not I, for a PlusPoints upgrade (although if I>0, PZ should also be >0). However, the reason that you were being charged here is that you had an upgrade that had already cleared. The rules do not allow you to go back after your Polaris flight has cleared and extend the request to your connecting flight. One person has reported repeated success doing that anyway, but the rules are quite clear -- if you request PlusPoints multiple times, you will get charged multiple times.

(Also, I'd be surprised if you had a fare break at ORD on your flight, as nearly all UA TATL fares disallow end-on-end connections to domestic flights -- they want to be able to offer a sale fare on, e.g., ORD-LHR without offering it to the rest of the country too).

As for the actual rule -- it gets really complicated. If you have a stopover, you need multiple upgrade instruments. If you have a fare break that would not be a stopover if you were through-ticketed, you do not need multiple instruments. If you have a fare break that would be a stopover, but the departure is the same calendar day, YMMV, but I'd expect that the computer would price it as a single instrument. If you do not have a fare break, but you do have an overnight transfer, the computer will (IME) require two instruments even though it's a through-fare; there are reports of people successfully getting an agent to process this with a single instrument. (Example: AUS-SFO / overnight / SFO-HKG; it wanted two instruments even though it was an AUS-HKG through fare).

Oh, and incidentally, multi-city search and broken fares are only tangentially related. UA will sell a broken fare on a regular round-trip search, and it will sell a through fare on multi-city search. However, I do agree that occasionally, UA's pricing system seems more likely to find a broken fare when using multi-city search.

I had a an a MUC-ORD-HNL in December with an (< 23 hour) overnight connection, and only one set of PP got deducted for it. Although ORD-HNL did end up not clearing in the end, which was kind of expected.

< 24 hours should work with 1 set of +Ps, but sometimes you have to call to make it happen.