I just tried to use my Monzo in Incheon Airport and it didnāt work.
I didnāt order a new card before I travelled though as didnāt think itād be a problem!
I had this same problem in Starbucks at Incheon last year. Rika has responded above with:
we found a few issues specific to some terminals in South Korea that have been resolved in a new revision of our cards.
If youāre going again soon, you should order a new card.
Received new card before vacation
Tried whilst there no success but luckily revolut worked fine
I think (foreign) MasterCards per se have tendency to fail in SK. Where it fails, (Revolutās) Visa worked fine for me.
Had the same issue on Incheon (very embarassingly, had to go through 3 cards) with both Monzo and Halifax so I donāt think it is bank specific 
Iām inclined to agree, even my Canadian MasterCards often fail in SK. Diners Club/Discover seems to be most widely accepted as they are partnered with the domestic SK payment network.
My tips for South Korea would be donāt use it. It fails in 95% of cases, even with international brands such as Starbucks and Uniqlo. I was not at any point able to use an ATM. The only successes I had were at the airport.
Just came back from South Korea, and ATMs for Foreign Banks (Standard Charted Bank, Citibank) works like a charm. Whereas for the Local Banks, have tried to use Woori Bank ATMs at the Incheon airport but to no avail, so Shinhan Bank ATMs will be the best option and they are pretty readily available in the major touristy areas.