The use of indefinite kernels has attracted many research interests in recent years due to their flexibility. They do not possess the usual restrictions of being positive definite as in the traditional study of kernel methods. This paper introduces the indefinite unsupervised and semi-supervised learning in the framework of least squares support vector machines (LS-SVM). The analysis is provided for both unsupervised and semi-supervised models, i.e., Kernel Spectral Clustering (KSC) and Multi-Class Semi-Supervised Kernel Spectral Clustering (MSS-KSC). In indefinite KSC models one solves an eigenvalue problem whereas indefinite MSS-KSC finds the solution by solving a linear system of equations. For the proposed indefinite models, we give the feature space interpretation, which is theoretically important, especially for the scalability using Nyström approximation. Experimental results on several real-life datasets are given to illustrate the efficiency of the proposed indefinite kernel spectral learning.