Anytime it is "cold" air source heat pumps don't work. Cold varies a bit, somewhere between -5C and - 25C depending on design factors. Even in the best case as you get closer and closer to the minimum temperature the worse they work (IE when you need them most!), and once you hit the cut-off you better have a backup source off heat.

You can use geothermo (ground) to work around this. I'd recommend it, but the one time install costs mean it is questionable if it is cost effective.

OK, so that's one, very well understood case. Yes, a heat pump will not work all the time. I have one and it stops working around 20F and the furnace takes over. So what? I live at the US/Canada border and my furnace runs maybe ~2 months during the year. This is still tremendous savings.

Big savings, but is it big enough to be worth the extra expense of a heat pump vs used using the furnace year round. The times when the heat pump works are the times when you least need it, since other activities of life are adding heat to the house too.