How significant is a power cord upgrade?

FERRARI, good link.
I have a Pass Labs amplifier and I'll be honest, I have not found earth shattering improvements with an aftermarket power cord with this amp(power cords tried were up to $700).
However, it is not the case with ALL amplifiers and other components. I think the differences you hear or don't hear with upgraded power cords on the amps depend on the power supply design and quality.
For example, a $6 Volex power cord sounds excellent. A Shunyata Taipan Alpha Helix cord does sound just a bit better, but not what you would call a huge difference.
I had several McCormack amps, where an upgraded power cord always made bigger difference than I experienced with the Pass Labs amp. So I don't think that any general rule applies here.
It has to be determined on a specific amp with a specific power cable only by trying.

That being said, I know someone with a Primaluna integrated using a Shunyata Taipan Alpha cord and says it is a very nice improvement.

I take the power cords that come with PCs, put them in the freezer for one month. Then use them. Significant improvement

Interesting, because in my all Krell system, expensive power cables made a BIG difference over the stock cords. I'm happy. ;-) Just experiment and see. I'm glad I did.

".... I'm considering upgrading the power cord. How significant is it compared to upgrading speaker cords and interconnects? Just wondering if I'll notice the same positive effects."

IMO
1. yes
2. maybe

I feel the power end of things, as has been recognized above, is where to start messin' with wires.

Foundation. gotta get a good one... naturally, that's taking for granted good components are already in place.

thereafter, getting as good a speaker wires as is possible is next.

Finally, ICs. those last two moves are debateable... but I feel PCs are top o' the list accessories.

Results do vary with PCs, as with any legit component. Generally speaking, larger AWG on power amps is a good thing. though not an absolute.

I've tried a fair amount of pc's on all my components. Whenever I get one in to try, it makes the circuit. Why not? I've yet to try Tvads notion of the $7 cord, but likely will at some point. For ex. I tried a highly recommended Purist Venusta pc once for a couple weeks. Put it right onto my pre which is where all had said it would work well. Didn't care for it at all. Good for me. As at a grand a pop, I saved a grand right there.

Tastes and system syn play a big role in what folks believe is or are great matches or devices, that and philosophy.

the litmus test is the Audiogone "Buy & Try shuffle", or if fortune is your's, borrow and listen. Then, there's the "Rent it and see" bit, too.

if the trepidation is are wires smake oil... myths... or a waste of funds, I'd say given my experience with them, "No. They impact a system as much as any other aspect of it."

Wires and isolation and as Fishboat said above, 'room treatment', can also embelish, detract, tame, or give life to the experience of home audio.

it all matters. how much, and how much you wish to spend to get how much, is really the answer. Great components alone, without some delving into what peeripherals can do for them is like putting recaps on a Jaguar. you'll likely not see what they are really capable of if after market (OEM) wires aren't investigated.

Upgrading PCs doesn't always guarrantee the positive improvement. The PS audio PCs chokes my 400xi and the 4 bsst amp. (sibilant/ bright HQ) but it works on the CDP with some improvements for better details. I went back to the stock PCs, both amps sound good as used to. PCs make difference but not necessary for the better.