This paper describes the use of single-scale measures to determine the level of randomness and complexity of a sequence. Such sequences originate from either various pseudorandom number generators or natural sources of white and coloured broadband noise. The paper provides a study of seven classes of sequences using the algorithmic complexity measures (the Kolmogorov-Chaitin complexity) and the probabilistic entropy-based measures (Shannon entropy). The study shows the fundamental differences between the two measures. The single-scale measures are adequate to determine the relative randomness and complexity of a sequence. However, they are not capable of revealing the hidden patterns in scale-invariant (self-affine) sequences. This paper identifies the need for new measures for such self-affine stochastic and chaotic sequences, and investigates if the existing techniques could be modified for the multiscale measures.