Ernest Kummer's work on Fermat's last theorem (FLT) basically forms the starting point of algebraic number theory. Although FLT was finally settled by very different means, the classical approaches are still interesting to look at. We will look at the approaches to the first case of FLT, by Kummer, Mirimanoff and others. Then we look at the more general problem of singular integers in cyclotomic fields. The older methods, together with a method using the p-adic Iwasawa logarithm, can be used to give some information about these singular integers.