Avoided level-crossing muon spin resonance (ALC-μSR) has been used to study the dynamics and local environment of spin probes formed by muonium (Mu) addition to 2-phenylethanol (PEA) and limonene (1-methyl-4-(1-methylethenyl)-cyclohexene) in an aqueous dispersion of the nonionic surfactant C₁₂E₄ (tetra(ethylene glycol) n-dodecyl ether). The spin probes derived from both cosurfactants reside within the micelles in the L₁ phase and the bilayers in the L_α phase rather than in the aqueous region. The local polarity measured by the different isomers of the Mu adducts of PEA suggests there is a water gradient within the micelles and bilayers. Slow rotation of the micelles broadened the Δ₁ resonances with increasing temperature in the L₁ phase while narrower Δ₁ resonances were observed in the L_α phase due to the rapid rotation of the spin probes around a preferred axis, which was wobbling within a cone.