We study the persistence properties of the [Formula presented] coarsening dynamics of one-dimensional q-state Potts model using a modified mean-field approximation (MMFA). In this approximation, the spatial correlations between the interfaces separating spins with different Potts states is ignored, but the correct time dependence of the mean density [Formula presented] of persistent spins is imposed. For this model, it is known that [Formula presented] follows a power-law decay with time, [Formula presented] where [Formula presented] is the q-dependent persistence exponent. We study the spatial structure of the persistent region within the MMFA. We show that the persistent site pair correlation function [Formula presented] has the scaling form [Formula presented] for all values of the persistence exponent [Formula presented] The scaling function has the limiting behavior [Formula presented] [Formula presented] and [Formula presented] [Formula presented] We then show within the independent interval approximation (IIA) that the distribution [Formula presented] of separation k between two consecutive persistent spins at time t has the asymptotic scaling form [Formula presented] where the dynamical exponent has the form [Formula presented] The behavior of the scaling function for large and small values of the arguments is found analytically. We find that for small separations [Formula presented] where [Formula presented] while for large separations [Formula presented] [Formula presented] decays exponentially with x. The unusual dynamical scaling form and the behavior of the scaling function is supported by numerical simulations. © 2003 The American Physical Society.