The original purpose for the generalised quadrature rules was to tackle irregular oscillatory integrands on a finite interval [a, b]. As the work progressed it was clear that the same approach could be applied to infinite range and singular integrals. This paper deals with applications to infinite range integrals, with particular emphasis on non-exponential integrands (which make classical Laguerre less easy to apply effectively). The crucial feature to allow these integrals to be evaluated was to replace the usual cosine weighted abscissae (used in the irregular oscillatory case) with a geometric distribution, or use the abscissae {1/xj}. A third set of abscissae were obtained from transforming the cosine weighted points onto [0,inf]. Comparisons of these choices on a range of practical integrals completes the paper. The second class of singular integrals appears in a related publication not presented for the current exercise. This work extends generalised quadrature methods to infinite range problems which makes the method applicable to a much wider class of integrals.