When to hand notice in?

[Lulu44]

Sorry if this has been asked before.
I've recently accepted a new job offer, I've completed various forms and submitted an online Dbs check yesterday, and the new employer is requesting references.
My current employer requires 1 months notice and ideally I'd like to start the new job ASAP (much better money and currently skint).
Do I wait a week or two and hand my notice in, or play it safe and wait for dbs and references to come back first?

Also, should I verbally notify my supervisor that I've got a job and should be handing in my notice in the near future, or leave it until I do hand it in?

Thanks in advance

Watching as I recently asked this and I think i did it the wrong way around.
I handed my notice in when all other checks were complete and The only outstanding one was the reference from my current job. I thought it bad form to do it the other way around.
I was really worried though in case the reference came back with something that they new company didn't like and company withdrew the offer, and my notice had been accepted.
Thankfully all was well.