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Here's the response I got from RC on Friday, so it doesn't look like they'll be rolling the faulty update back this time.

The issue that you have reported in this ticket is related to a Known Issue we currently have in our system.

At this point, we will be placing this case into a PKI Hold status. PKI stands for Product Known Issue. This means that we are investigating it as a bug on our side and there is no currently available ETA when it will be resolved. We have reproduced the issue and are working on resolving it. You will not receive much in the way of correspondence while they resolve this issue, however, you will be notified when the issue is fixed.

Yes, we've had the same faulty update applied again this morning too. Though this time we can transfer calls, but adding a call to create a conference or converting to a video call are greyed out.

We're a new customer. Is this sort of thing common? How often can we expect these sort of breaking changes?

Don't worry, it's very rare. We've been using RC for around 18 months now and this is the first time they've pushed out an update that has created a clear fault. They're normally very reliable.