Constructing divisors with prescribed singularities is one of the most powerful techniques in modern projective geometry, leading to proofs of major results in the minimal model program and the strongest general positivity theorems by Angehrn-Siu and Kollár-Matsusaka. We present a novel method for constructing singular divisors on surfaces based on infinitesimal Newton-Okounkov bodies. As an application of our machinery we discuss a Reider-type theorem for higher syzygies on abelian varieties building on earlier work of Lazarsfeld-Pareschi-Popa.