We apply Bayesian variable selection to investigate linear factor asset pricing models for a large set of candidate factors identified in the literature. We extract model and factor posterior probabilities from thousands of individual stocks via Markov Chain Monte Carlo estimation together with the exact distribution of pricing statistics. Our results show that only a small number of factors are relevant and, except for the market and size factors, these are not the factors in widely used linear factor models such as Fama and French (2015, Journal of Financial Economics 116, 1–22) or Hou et al. (2015, The Review of Financial Studies 28, 650–705). Moreover, many different linear factor models achieve similar empirical performance, suggesting that the search for a single linear factor model is unlikely to yield a definitive answer.