MacBook Air keyboard shows a pound sign (as Shift 3), but when I type this, I get a hash sign.

lbomaak2 wrote:

2. Check that the keyboard setting is UK English rather than US English. I have done this, and the keyboard is definitely set to UK English.

There is no keyboard called UK English, so you may be looking at the wrong setting. You need to have the Input Source set to British to have £ on shift 3.

To do this you go to system settings > keyboard > text input > edit > input sources and remove everything except British.

lbomaak2 wrote:

OK, so I have got both British and Unicode Hex input. How do I remove the Unicode Hex input?

in the input source settings, you select it and click on the minus button

lbomaak2 wrote:

1. Use Option 3 rather than Shift 3; but when I do this, no character appears on the screen at all (and I have also tried Shift+Option 3, with the same result).

That is certainly not normal, some character should be produced. Does that happen in all apps? Do you get something when you do option 12456 etc? Do you have the Unicode Hex input source active? If so, remove it.

lbomaak2 wrote:

1. On my MacBook Air, the keyboard shows a pound sign (as Shift 3), but when I type this, I get a hash sign. This appears to be a common issue, and I have tried both methods recommended to overcome it:

Use Option 3 rather than Shift 3; but when I do this, no character appears on the screen at all (and I have also tried Shift+Option 3, with the same result).

2. Check that the keyboard setting is UK English rather than US English. I have done this, and the keyboard is definitely set to UK English.

Any other ideas?

By "pound sign", do you mean the sign that indicates the UK currency? I only ask because, in the U.S. both "pound" and "hash" refer to the octothorpe (#). And, on the English (U.S.) keyboard, shift-3 produces #.

What language(s) do you have selected in Settings>General>Language & Region?

shift-3: #

Option-3: £

Apparently, option-3 isn't working for the OP.

The cavalry has arrived!