For travelers in Czech Republic, how common is it to be able to pay using a card (Visa/Mastercard)?
Can you pay for public transport tickets by card? Can you generally pay in restaurants by card? Is it possible to use non-standard cards, such as American Express or Discovery?
Payments cards are very popular and all major retailers accept them. Smaller shops and pubs are reluctant to accept it because of the fees so you should watch sign at doors. The smaller city is the bigger is chance that cards will be not accepted. But ATMs are quite common. Here is a map of ATMs: http://www.kurzy.cz/banky/bankomaty/
The Czechia uses EMV standard as all Europe so only chip/contactless cards are issued. It may happen that your card will ask you for a PIN so do not be surprised. Or the transaction can be declined if the terminal insists on PIN and your card cannot handle it. Though I think that signature card holder verification method will work in most cases. Clerk will not be used to as local cards do not have fallback to magnetic stripe.
Other schemes are troublesome. VISA, VISA Electron, MasterCard and Maestro are supported everywhere. V-Pay is usally accepted as well. Amex and diners are accepted rarely, especially at taxi, hotels or frequent tourist places.
As for the public transport, this differs city by city. In Prague, most of the ticket machines only accept coins, but the major hubs (e.g. at airport, major train stations) are recently being outfitted with ticket machines that also accept payment cards (and banknotes). However I noticed some of them only accept contact-less cards.
Major train stations all accept payment cards.
Virtually all supermarkets and most of shops accept payment cards in larger cities. Even in smaller ones, supermarkets will often accept payment cards.
As for types of cards accepted - almost all domestic payment cards are issued by Visa or MasterCard, so you won't have problems with those. Don't know about others. The accepted authorization methods are contact-less, chip-and-pin, often magnetic stripe and pin should be possible. In the past, magnetic stripe and pin or signature were used, so many terminals still have the stripe reader. But I wouldn't rely on signature being accepted instead of pin anymore.
In Prague, cards were accepted many places except, curiously, the apartment hotel we used. All restaurants above street food, all souvenir shops, all museums, all major stores. American Express was accepted in a few places, including a special price at the cinema that outweighed their outrageous 2.75% foreign currency fee. Discover was also accepted at a few stores, including an indulgent purchase of a tea set, for which the 0% conversion fee and 2% end-of-year rebate were appreciated.
On an earlier trip (2010), I had trouble finding ATMs in the countryside and restaurants did not accept plastic.
