This paper presents three novel methods for the multiobjective design of controllers with tunable parameters and fixed structure for linear systems. These techniques exploit the geometric properties of varieties and envelopes to obtain a closed-form expression of the set of candidate Pareto optimal values. The design objectives span from the regional pole placement to more general quadratic cost functionals, thus tackling a wide variety of control tasks. Examples of applications of the proposed techniques both to (possibly, nonconvex) classical benchmark problems and to physical plants are given throughout the paper.