We explore the use of various element-based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis. The rules studied encompass tensor-product Gauss and Gauss–Lobatto rules, and certain so-called monomial rules that do no possess a tensor-product structure. The objective of the study is to determine quadrature strategies, which enjoy the same accuracy and stability behavior as full Gauss quadrature, but with significantly fewer quadrature points. Several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard C0-continuous finite elements of the same order. •We test various element-based reduced quadrature rules for quadratic and cubic spline elements.•They encompass tensor-product Gauss and Gauss–Lobatto rules, and monomial rules.•Some rules enjoy the same accuracy and stability as full Gauss quadrature, but with significantly fewer quadrature points.•They can substantially reduce the formation and assembly effort in isogeometric analysis.