The paper starts with a brief review of some basic facts about Laguerre geometry. Then \(L\)-minimal canal surfaces are studied. \(L\)-minimal canal surfaces are enveloping surfaces of a 1-parameter family of oriented spheres that are extremals of the variational problem defined on immersed surfaces in Euclidean 3-space by the functional \((f, S)\mapsto \int_S (H^2- K)K^{- 1} dA\). Here \(H\), \(K\) and \(dA\) denote mean and Gauss curvature and the induced area element, respectively. Special conditions on the invariant functions are obtained and adapted coordinate systems are introduced. On these grounds, \(L\)-minimal canal surfaces are divided in two main types and six classes. Finally, explicit solutions for surfaces of both types are given.