The biggest scam, which most people have already fallen for is to believe daft boris and his cronies that ordinary people should all stay at home.
That said, the next scam is predicted to be with this bunch of, Serco run, so called contact tracers - "Hello Mr/Mrs Mug, you have been in contact with someone who has Koronaberg virus. Please give me your email address, so I can send you a form to fill in.......". [Then I'll have your email address etc. etc. to bombard you with spam and links to my very nice malware!]