The first year after opening a credit card, it is common for the cc to wave the annual fee (AF), but rare for them to give you 0% interest.
Chase Slate gives 0% for the first 18 months. (confirm this info)
Slate is generally 0% on balance transfers made with a limited time of opening the card.
There are plenty of 0% APR promo offers out there for new cards, and Discover will offer them on existing accounts occasionally.
Chase Slate, Chase Freedom, Chase Ink Cash all have 0% promo APR's. Citi Simplicity always has like 21 months on new accounts for 0% promo APR's. Several AMEX cards offer 0% promo APR's. BofA etc etc etc etc