Fixed multiword expressions are common in many, if not all, natural languages. In the Universal Dependenciesframework, UD, a subset of these expressions are modelled with the dependency relation fixed, targeting themost grammaticalized cases of functional multiword items. In this paper we perform a detailed analysis of 439expressions modelled with fixed in two Swedish UD treebanks in order to reduce their numbers and fit thedefinition of fixed better. We identify a large number of dimensions of variation for fixed multiword expressionsthat can be used for the purpose. We also point out several problematic aspects of the current UD approachto multiword expressions and discuss different alternative solutions for modelling fixed expresions. We suggestthat insights from Constructional Grammar (CxG) can help with a more systematic treatment of fixed expressions in UD.