In a lot of the markets our customers trade in, an outage is a weekly event rather than an incident. A POS that stops selling when the link drops is not a POS you can run a shop on, so offline operation is a base requirement here rather than a premium feature.
What continues offline: ringing up sales, printing receipts, taking payment. What waits for the connection: anything that by definition needs the server, such as cross-store stock visibility or a real-time fiscal submission where the tax authority requires one.
Worth planning for separately: a power cut is not the same as a network cut. If the area loses power regularly, the money question is a UPS on the till and the router, not software — the software will survive either way, but a computer that loses power mid-transaction is a different kind of risk.