Every receipt breaks out the VAT and rate. Food, books and children's clothes (and a few other things) are exempt (UK).
All online purchases show VAT in the checkout and basket, and in the invoice after purchase.
Computers can deal with the latter, signage can't.
Might be a little different for online stores where tax can't be calculated until they know which of the many different local tax codes the sale is being made under, but as you've already noted computers can handle updating the price before people confirm the purchase.
Except for businesses, there isn't the situation as in the US, where the retailer is responsible for verifying that a customer is tax-exempt. People on the equivalents of food stamps pay the VAT; yes the money just goes in a circle back to the government, but it's probably easier accounting for everyone.
The till receipt or invoice breaks down the VAT element for you, and most places even pick out what is zero rated or exempt should you care.