  Loops are subgraphs responsible for the multiplicity of paths going from one
to another generic node in a given network. In this paper we present an
analytic approach for the evaluation of the average number of loops in random
scale-free networks valid at fixed number of nodes N and for any length L of
the loops. We bring evidence that the most frequent loop size in a scale-free
network of N nodes is of the order of N like in random regular graphs while
small loops are more frequent when the second moment of the degree distribution
diverges. In particular, we find that finite loops of sizes larger than a
critical one almost surely pass from any node, thus casting some doubts on the
validity of the random tree approximation for the solution of lattice models on
these graphs. Moreover we show that Hamiltonian cycles are rare in random
scale-free networks and may fail to appear if the power-law exponent of the
degree distribution is close to 2 even for minimal connectivity grater than 3.
