Tried searching but was unsuccessful --

If I have a ticket which has both a refundable segment and a non-refundable segment, and I now need to cancel the trip, how does the refund work?

Entire PNR refundable?
Entire PNR non-refundable?
Refundable segments of PNR refundable and non-refundable segments of PNR not?

Unfortunately the more restrictive fare rules almost always are the rules that govern the entire PNR with different fare rules. So its highly likely the whole PNR is non-refundable.

Ah, so if I have a PNR that looks like this:

U to connection Y to destination and
Y to connection and A to origin

Assuming U is not refundable (it's not right?), the whole PNR would be non-refundable?

The determining factor is not whether both segments are in the same PNR but whether both segments are on the same ticket. If they are on the same ticket, then the whole ticket is nonrefundable. If they are in the same PNR but on different tickets, then the refundable segment is refundable and the nonrefundable segment is nonrefundable.

You have to check the fare rules. U is not usually refundable but it can be. A shortcut to reading the fare rules (but not foolproof) is to check the fare basis. If there is an N at the end like U7N then its not refundable. You may want to check the A fare rules also. Those aren't always refundable either.

This may be true, but in practice, won't all segments for the same passenger on the same PNR also be on the same ticket, save some extraordinary situation like there being too many segments for the same ticket? Put another way, one can assume that in most cases, same PNR = same ticket, and that the "most restrictive fare rule applies to all segments" rule is applicable.

And yes, this can result in some unusual situations, which sometimes can seem very unfair. For example, you buy a $10,000 J fare with no restrictions, and add on a $100 Q fare to make a short domestic connection, and the Q fare has a change fee and is non-refundable. That makes the $10,000 J fare also have a change fee and also be non-refundable.

If you're buying your tickets online, then that is generally true, but a travel agent can put multiple tickets on the same PNR (depending on carrier rules).

The answer is actually more nuanced than everyone is suggesting because UA is a bit more generous with its fare rules for itineraries that combine more than one fare component than many airlines are ... note the exception in the following provision (or one like it) in most of United's fare rules under both the cancellations and changes sections:
The majority of domestic roundtrip fares are combined end-on-end because they are essentially a combination of two one-way fares, even if they are on a single ticket. This is generally NOT true of international fares (which are almost always either roundtrip fares or two fares combined on a roundtrip basis). Accordingly, the unrestricted fare components may be changed without paying a change fee (although it may be a challenge to find someone to do this as, as far as I can tell, the tools that United uses won't recognize this automatically, so you would have to find an agent that can read and understand the fare rules, then be willing to manually reissue the ticket). This probably wouldn't apply to an actual refund to your original form of payment, but you should be able to at least get travel credit for the full amount of the refundable part of the ticket without paying any sort of fee.

I was thinking about this and isn't it only really useful if you don't have to change/cancel the non-refundable portion? Otherwise you still have to pay the change fee. Or I guess it can also be helpful if the non-refundable segment is worth less than the change fee.

You are correct ... though given that the language appears in the "cancellations" section, it is possible (although not particularly likely) the OP may be able to get a refund for the refundable segments and travel credit (after change fee) for the nonrefundable segments if he finds an agent in a generous mood.

Thanks for the detailed answers, all!