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Motivated by the recently measured muon's anomalous magnetic moment$a_{\mu}$, we examine the supersymmetry contribution to $a_{\mu}$ in variousmediation models of supersymmetry breaking which lead to predictive flavorconserving soft parameters at high energy scale. The studied models includedilaton/modulus-mediated models in heterotic string/$M$ theory, gauge-mediatedmodel, no-scale or gaugino-mediated model, and also the minimal and deflectedanomaly-mediated models. For each model, the range of $a^{SUSY}_{\mu}$ allowedby other experimental constraints, e.g. b --> s\gamma and the collider boundson superparticle masses, is obtained together with the corresponding parameterregion of the model. Gauge-mediated models with low messenger scale can giveany $a^{SUSY}_{\mu}$ within the $2\sigma$ bound. In many other models, b -->s\gamma favors $a^{SUSY}_{\mu}$ smaller than either the $-1\sigma$ value($26\times 10^{-10}$) or the central value ($42\times 10^{-10}$).Comment: RevTeX, 29 pages, 14 eps figures, figure for deflected anomaly  mediation is corrected, reference adde

Probing the messenger of supersymmetry breaking by the muon anomalous magnetic moment