A Newton–Okounkov body is a convex body associated to choices of a big line bundle and a complete flag of subvarieties of a variety. A basic assumption in the usual (Lazarsfeld-Mustata) construction of the Newton-Okounkov body is that the point of the flag is a smooth point of each piece of the flag. Unfortunately, a general family of flags will have flags with one of the subvarieties singular at the point as soon as the dimension of the base of the family is large enough an inconvenience when one tries to study the variation of Newton-Okounkov bodies with the flag. We propose a generalization of the Newton–Okounkov body associated to a big line bundle on a smooth surface with respect to a flag containing a possibly singular point, which recovers the usual definition when the point of the flag is a smooth point of each subvariety in the flag, and discuss some results in this direction.