We consider the problems of computing the maximal and the minimal non-empty suffixes of substrings of a longer text of length n. For the minimal suffix problem we show that for every tau, 1 <= tau <= logn, there exists a linear-space data structure with O(tau) query time and O(n logn/tau) preprocessing time. As a sample application, we show that this data structure can be used to compute the Lyndon decomposition of any substring of the text in O(k tau) time, where k is the number of distinct factors in the decomposition. For the maximal suffix problem, we give a linear-space structure with O(1) query time and O(n) preprocessing time. In other words, we simultaneously achieve both the optimal query time and the optimal construction time. (C) 2015 Elsevier B.V. All rights reserved.