  We investigate the dimensional crossover of scaling properties of avalanches
(domain-wall jumps) in a single-interface model, used for the description of
Barkhausen noise in disordered magnets. By varying the transverse aspect ratio
$A=L_y/L_x$ of simulated samples, the system dimensionality changes from two to
three. We find that perturbing away from $d=2$ is a relevant field. The
exponent $\tau$ characterizing the power-law scaling of avalanche distributions
varies between $1.06(1)$ for $d=2$ and $1.275(15)$ for $d=3$, according to a
crossover function $f(x)$, $x \equiv (L_x^{-1})^{\phi}/A$, with $\phi=0.95(3)$.
We discuss the possible relevance of our results to the interpretation of
thin-film measurements of Barkhausen noise. We also study the probability
distributions of interface roughness, sampled among successive equilibrium
configurations in the Barkhausen noise regime. Attempts to fit our data to the
class of universality distributions associated to $1/f^\alpha$ noise give
$\alpha \simeq 1-1.1$ for $d=2$ and 3 (provided that suitable boundary
conditions are used in the latter case).
