There's been debate through the years, and some users have even run some
challenges to try, and prove that they can be secure. However, generally brain wallets introduce additional risks, which if you aren't aware of could potentially compromise you. Plus, as suggested humans are generally absolutely horrible at entropy. We tend to think similarly, and follow patterns. It's very much what makes us human. Most living things have patterns.
The above challenge was never claimed, however this is a rather flawed challenge since we don't know exactly how it was generated. The question isn't about are brain wallets inherently insecure, it's more the human factor when it comes to generating the entropy. LoyceV, has a nice little thread there that demonstrates this weakness.
So, even if you believe you've made a completely random password, it likely isn't. Our thought processes are flawed for entropy. Computer generation of entropy is almost always better.
Thank you for pointing out this thread from CIYAM, it seems the challenge is over since the last funds have been spent in 2019 but at least it shows one thing. (Strong) brain wallets could be used for a temporary storage.
If you need to go somewhere where you can't safely bring any luggage with you, it could be a solution. I know it's certainly safer to store a seed on internet and to memorize its passphrase in this kind of situation since the attacker would have to find and hack your account first. But if you cant easily access internet in this place or if you don't want to rely on the availability of an online service, only few solutions remain...
Another famous use case is a way to easily and quickly give funds to someone you trust.
