Monte Carlo (MC) analysis of the Goldstone mode singularities for the transverse and the longitudinal correlation functions, behaving as G_{perp}(k) simeq ak^{-lambda_{perp}} and G_{parallel}(k) simeq bk^{-lambda_{parallel}} in the ordered phase at k -> 0, is performed in the three-dimensional O(n) models with n=2, 4, 10. Our aim is to test some challenging theoretical predictions, according to which the exponents lambda_{perp} and lambda_{parallel} are non-trivial (3/2<lambda_{perp}<2 and 0<lambda_{parallel}<1 in three dimensions) and the ratio bM^2/a^2 (where M is a spontaneous magnetization) is universal. The trivial standard-theoretical values are lambda_{perp}=2 and lambda_{parallel}=1. Our earlier MC analysis gives lambda_{perp}=1.955 pm 0.020 and lambda_{parallel} about 0.9 for the O(4) model. A recent MC estimation of lambda_{parallel}, assuming corrections to scaling of the standard theory, yields lambda_{parallel} = 0.69 pm 0.10 for the O(2) model. Currently, we have performed a similar MC estimation for the O(10) model, yielding lambda_{perp} = 1.9723(90). We have observed that the plot of the effective transverse exponent for the O(4) model is systematically shifted down with respect to the same plot for the O(10) model by Delta lambda_{perp} = 0.0121(52). It is consistent with the idea that 2-lambda_{perp} decreases for large $n$ and tends to zero at n -> infty. We have also verified and confirmed the expected universality of bM^2/a^2 for the O(4) model, where simulations at two different temperatures (couplings) have been performed.
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