When moving what’s expected

Are you moving out of a rental? Or from a house you have sold?

Rental - It's somebody else property. You need to repair (will be in lease).

Selling - Caveat Emptor. The house is sold as is.

Unless your contract says otherwise, fixtures and fittings are included in the sale and should be left in the house. Anything that's bolted to the wall looks a lot like a fixture and, strictly speaking, should not be removed unless you have express agreement for the removal.

Having said that, it's unlikely that the purchaser wants your headboard, so he will probably readily agree to its removal. But when you're having that conversation, that's the time to deal with the issue of damage to the wall. I think it would be reasonble to offer to fill the hole and touch up the paintwork. If I were the purchaser, I would probably make my agreement to the removal conditional on the vendor making good the wall (unless I was planning immediate work that would fix the problem anyway).

Maybe leave the rail too? It's a fixture. Plus, that way the new owner can decide whether he wants holes in the wall or not.

Obviously, you can stipulate in your contract of sale that you are removing certain fixtures and fittings. No problem there. This is quite commonly done. Be sure to put the lights in the contract when you sell your house!

(Seriously. I know of a dispute between a seller who removed a built-in fish tank and a buyer for whom the built-in fish tank was a major attraction that influenced his decision to buy the house. Neither had thought to mention to the other the importance of the fish tank to them, and the contract did not refer to it.)

As to what exactly amounts to a fixture: a lot of Ikea furniture is fixed to the wall to prevent it from falling over — there's a screw in the wall and a bracket or a strip of fabric that is in turn secured to the furniture. The screw and the bracket are fixtures; the furniture, probably not. A picture rail is a fixture and, by analogy, I'd say that a strip of wood or metal that serves as the base of a French cleat, as described by ted1, is a fixture.

In general, if it's fixed to the fabric of the house and if removing it would damage the fabric in way that would require repair, it's a fixture. But some things that don't tick those boxes are still considered fixtures — e.g. fitted carpets.

Leave them be, buyer won't care.