One possibility I'm excited about is users being able to perform computations locally without sending their data anywhere, and then providing the results to a company, government, etc with a proof that the results are faithful.
(1) This is my credit score, certified by XYZ agency, so please don't ask for my SSN so that you can lose it in a public database leak tmrw
(2) Here's a bug in your program, please give me the bug bounty and I will tell you the bug (can help stop sketchy bug bounty programs.)
(3) Your Certificate Transparency Provider can prove that, for the latest root, there was no change in your certificate. (This has less to do with privacy and more to do with the succinct verification properties of the latest zkps)
(4) Construct postquantum-secure signatures (eg: the Picnic signature scheme)
Generally, ZKPs provide selective disclosure: I can prove to you that some fact about me or my accounts is true, without revealing to you any other information. The SSN example is one, you could generalize that to taxes, bank statements, Keybase attestations, etc.
3 seems cool but not particularly impactful in terms of the guarantees offered by CT.
4 can be achieved in many ways without ZKPs.
(2) As the bug bounty claimant, I compile the program into the zkp proof, run the program with the bad input that leads to the exploited state, and submit the witness to the code author that proves that I know of an input that causes the code to reach an undesirable state (without revealing it).
You could prove you're of legal drinking age without having to show a stranger your ID full of sensitive info.
You could prove you "liked" a band before a certain date and therefore get discounts on merch or show tickets or something.
New ideas come to me every day.
I worked with the Brave team to sketch out how the latter could be done in their system. It's 1 of 10 proposals and iirc half are using ZKPs to reduce information given to advertisers.