North Korea: Like Vietnam, “formerly”. North Korean vocabulary shows a tendency to prefer Native Korean over Sino-Korean or foreign loanwords, especially with recent political objectives aimed at eliminating foreign influences on the Korean language in the North. Ideally, they wish to completely abolish the usage of foreign loanwords that includes Chinese Characters.
In the early years, the North Korean government tried to eliminate Sino-Korean words and Hanja (“Chinese Characters”), however, it failed due to the majority of the vocabulary and etymology derived from Chinese. The only successful part about this attempt was the abolishment of Hanja.
South Korea: Depends on who you are talking about? The older generation like my grandparents is almost definite. From my parents to my generation, it slowly transitions from “mandatory” to “voluntary” and eventually going to be a “hit-or-miss” like Latin class course in the U.S. (met some who took the class and some who never took the class). If it's me you're talking about, I do use it to benefit other languages such as Chinese, Japanese, and surprisingly Vietnamese. Overall, I would have to say out of the entire Korean population, it would be “occasionally”.
Here are some Chinese Characters you can find in Korea, nowadays.
One of the most common ones is the “newspapers”, “articles” especially “online articles” or “news”
Notice that Chinese Characters is in Traditional Chinese Characters, Korean Hanja uses Traditional Chinese Characters.
[These Chinese Characters are abbreviated such as “日” (일): JPN, “中” (중): CHI, “美” (미): USA, “韓” (한): KOR, and “北” (북): DPRK(orea)]
Another is formal documents
A birth certificate.
Marriage certificate
Some ID Cards (formerly driver’s license and passport)
My grandfather's passport.
Signs or billboards (commonly in pagoda temples/ shrines but rarely in urban areas)
Many are in Chinese mostly because of Simplified Chinese (Korean Hanja doesn’t use Simplified Chinese). The only sign that IS Hanja/ Traditional Chinese Characters along with Sino-Korean is “군래” (君來).
Airports and subway station.