  Using an elaborate set of simulational tools and statistically optimized
methods of data analysis we investigate the scaling behavior of the correlation
lengths of three-dimensional classical O($n$) spin models. Considering
three-dimensional slabs $S^1\times S^1\times\mathbb{R}$, the results over a
wide range of $n$ indicate the validity of special scaling relations involving
universal amplitude ratios that are analogous to results of conformal field
theory for two-dimensional systems. A striking mismatch of the $n\to\infty$
extrapolation of these simulations against analytical calculations is traced
back to a breakdown of the identification of this limit with the spherical
model.
