These are all good indexes of monotonic association. Spearman's $\rho$ is related to the probability of majority concordance among random triplets of observations, and $\tau$ and $\gamma$ are related to pairwise concordance. The main decision to make in choosing $\gamma$ vs. $\tau$ is whether you want to penalize for ties in $X$ and/or $Y$. $\gamma$ does not penalize for ties in either, so that a comparison of the predictive ability of $X_{1}$ and $X_{2}$ in predicting $Y$ will not reward one of the $X$s for being more continuous. This lack of reward makes it a bit inconsistent with model-based likelihood ratio tests. An $X$ that is heavily tied (say a binary $X$) can have high $\gamma$.

These are all good indexes of monotonic association. Spearman's $\rho$ is related to the probability of majority concordance among random triplets of observations, and $\tau$ (Kendall) and $\gamma$ (Goodman-Kruskal) are related to pairwise concordance. The main decision to make in choosing $\gamma$ vs. $\tau$ is whether you want to penalize for ties in $X$ and/or $Y$. $\gamma$ does not penalize for ties in either, so that a comparison of the predictive ability of $X_{1}$ and $X_{2}$ in predicting $Y$ will not reward one of the $X$s for being more continuous. This lack of reward makes it a bit inconsistent with model-based likelihood ratio tests. An $X$ that is heavily tied (say a binary $X$) can have high $\gamma$.