In fault-line estimation in spatial problems it is sometimes possible to choose design points sequentially, by working one's way gradually through the "response plane," rather than distributing design points across the plane prior to conducting statistical analysis. For example, when estimating a change line in the concentration of resources on or under the sea bed, individual measurements can be particularly expensive to make. In such cases, sequential, design-adaptive methods are attractive. Appropriate methodology is largely lacking, however, and the potential advantages of taking a sequential approach are unclear. In the present paper we address both these problems. We suggest a methodology based on "sequential refinement with reassessment" that relies upon assessing the correctness of each sequential result, and reappraising previous results if significance tests show that there is reason for concern. We focus part of our attention on univariate problems, and we show how methods for the spatial case can be constructed from univariate ones.