Recently singular solutions have been discovered in purely elongational flows of visco-elastic fluids. We surmise that these solutions; are the mathematical structures underlying the so-called birefringent strands seen experimentally. In order to facilitate future experimental studies of these we derive a number of asymptotic results for the scaling of the width and extension of the near-singular structures in the FENE-P model for polymers of finite extensibility. (c) 2008 Elsevier B.V. All rights reserved.