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Oscillons are long-lived, spatially localized field configurations, which are supported by attractive non-linearities in the scalar potential. We study oscillons comprised of multiple interacting fields, each having an identical potential with quadratic, quartic and sextic terms. We consider quartic interaction terms of either attractive or repulsive nature. In the two-field case, we construct semi-analytical oscillon profiles for different values of the potential parameters and coupling strength using the two-timing small-amplitude formalism. We use analytical and numerical techniques to explore the basin of attraction of stable oscillon solutions and show that, depending on the initial perturbation size, unstable oscillons can either completely disperse or relax to the closest stable configuration. We generalize our analysis to multifield oscillons and show that the governing equations for their shape and stability can be mapped to the ones arising in the two-field case. Finally, we study the emergence of multicomponent oscillons in one and three spatial dimensions, both numerically and through Floquet theory.