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In recent years there has been an increasing interest in the use of relational event models for dynamic social network analysis. The basis of these models is the concept of an "event", defined as a triplet of time, sender, and receiver of some social interaction. The key question that relational event models aim to answer is what drives social interactions among actors. Researchers often consider a very large number of predictors in their studies (including exogenous variables, endogenous network effects, and various interaction effects). The problem is however that employing an excessive number of effects may lead to model overfitting and inflated Type-I error rates. Consequently, the fitted model can easily become overly complex and the implied social interaction behavior becomes difficult to interpret. A potential solution to this problem is to apply Bayesian regularization using shrinkage priors. In this paper, we propose Bayesian regularization methods for relational event models using four different priors: a flat prior model with no shrinkage effect, a ridge estimator with a normal prior, a Bayesian lasso with a Laplace prior, and a horseshoe estimator with a numerically constructed prior that has an asymptote at zero. We develop and use these models for both an actor-oriented relational event model and a dyad-oriented relational event model. We show how to apply Bayesian regularization methods for these models and provide insights about which method works best and guidelines how to apply them in practice. Our results show that shrinkage priors can reduce Type-I errors while keeping reasonably high predictive performance and yielding parsimonious models to explain social network behavior.