Researchers have used many different methods to detect the possibility of long-term dependence in stock market returns and, generally, there is mixed evidence for the presence of long memory in these data. In this paper, three different tests, (namely Rescaled Range (R/S), its modified form, and GPH ), in addition to a new approach using the discrete wavelet transform, (DWT), have been applied to the daily returns of five Irish Stock Exchange (ISEQ) indices. These methods have also been applied to the volatility measures ( namely absolute and squared returns). The aim is to investigate the existence of long-term memory properties. The indices are Overall, Financial, General, Small Cap and ITEQ and the results of these approaches show that there is no evidence of long-range dependence in the returns themselves, but there is strong evidence for such dependence in the squared and absolute returns. Moreover, the discrete wavelet transform (DWT) has the additional advantage of providing an in-depth view of the data sets and this gives us a real indication of structure in long memory effects e.g. giving clear picture of the movements in the series.