I'm using Safari on an iPhone 6, iOS 8.4 (yes, it's time to upgrade, but then how would we find these fun bugs?).
In the new mobile chat, the bottom part of the content area is often outside the bounds of the device and can't be scrolled to.
This is what I normally see when opening a room list: https://i.sstatic.net/QXRxx.png. And this is what happens if I take a screenshot while still touching the screen after scrolling to the bottom of the list: https://i.sstatic.net/s2NoZ.png. (The second I let go, the list "bounces" back to what you see in the first screenshot.)
Entering a room typically looks like this: https://i.sstatic.net/Lbt7v.png.
So, it appears that this behaviour occurs when Safari is still showing the browser navigation controls. There's no apparent way to dismiss those controls - scrolling or tapping anywhere on the screen doesn't do it - and they don't disappear on their own. I've been able to make it go away sometimes by scrolling until the input message box is in view, but then the top part of the page goes AWOL and there's no bringing it back: https://i.sstatic.net/V91VN.png.
If I reply to a specific message, the input box at the bottom sticks around like it should, and the messages scroll fine behind it... but the header disappears completely (similar to this screenshot, but without even a sliver of blue at the top).
