  A universal geometric cluster algebra over an exchange matrix B is a
universal object in the category of geometric cluster algebras over B related
by coefficient specializations. (Following an earlier paper on universal
geometric cluster algebras, we broaden the definition of geometric cluster
algebras relative to the definition originally given Fomin and Zelevinsky.) The
universal objects are closely related to a fan F_B called the mutation fan for
B. In this paper, we consider universal geometric cluster algebras and mutation
fans for cluster algebras arising from marked surfaces. We identify two crucial
properties of marked surfaces: The Curve Separation Property and the Null
Tangle Property. The latter property implies the former. We prove the Curve
Separation Property for all marked surfaces except once-punctured surfaces
without boundary components, and as a result we obtain a construction of the
rational part of F_B for these surfaces. We prove the Null Tangle Property for
a smaller family of surfaces, and use it to construct universal geometric
coefficients for these surfaces.
