Some mansard roofs have a vertical face that bulges outward in an "S" or bell shape. This creates more internal volume inside the attic. It is often seen in North America on the tops of the towers that crown many mansions of the Victorian Era. (See Reference 2) These houses may be Second Empire style, or may also be Italianate, Queen Anne, Romanesque, or High-Victorian Gothic. They borrowed the mansard roof from French architecture.
Concave mansard roofs have vertical faces that curve inwards or flare out at the bottom. This allows for dormer windows to protrude past the vertical roofline without extending past the front face of the building. (See Reference 2) This is the most common type of mansard roof and is regularly seen on 19th and early 20th century homes and on some older commercial buildings.
Some mansard roofs have no curve to their face at all, (See Reference 2) instead just having a very steeply angled flat face that may be peppered with windows. This is commonly seen on 1960s suburban houses that revived the mansard style, as well as on some older homes and downtown commercial buildings.