You can't force any shop or service provider to comply with their tax obligations.
Paying by cheque, BACS or card is more likely to ensure they declare the correct profits and pay the right tax, but still no guarantee.
Likewise, always asking for a receipt/invoice is more likely to ensure compliance, but again, still no guarantee.
I'm been an accountant for nearly 40 years, and dealt with hundreds of tax investigations. "Cash" related tax avoidance was huge but generally not from small scale transactions like your hairdresser or dance instructor which are usually pretty easy for the tax inspector to prove due to diaries, work sheets etc.
The really big cases were the likes of scaffolders and roofers, who'd do "cash in hand" jobs worth a few thousand. Likewise with "scrap" dealers who often deal in thousands of pounds of cash when buying/selling scrap metal, used tyres, etc.
What I can't understand is that despite this happening for decades, HMRC still aren't on top of it. Earlier this year, we needed scaffolding all around our house - quotes were typically £2k-£3k. We got around 5 or 6 quotes, all from reputable, well known, local firms, and nearly all said they'd knock off the VAT for cash with no receipt!
That's a lot bigger loss to the Treasury than a dance instructor "forgetting" to put £50 of cash takings through their books!
But as others have said, regardless of how you pay, get a written receipt/invoice to prove you've paid. That's more for your protection (in case they come back and say you didn't) than to prove they are paying tax on it.
For people with genuine suspicions about tax evasion, you can anonymously report it via the HMRC website.