I'm far from an expert on this subject, but the book "Higher-Dimensional Categories: an illustrated guide book" by Cheng and Lauda (available here) has a very nice description of the things that various competing notions of n-categories are trying to generalize and the motivations behind them. It's not quite a cut-and-dried "list of properties", but my impression is that the philosophy behind the various definitions is too complicated to just make a single list.

I'm far from an expert on this subject, but the book "Higher-Dimensional Categories: an illustrated guide book" by Cheng and Lauda (available here) has a very nice description of the things that various competing notions of n-categories are trying to generalize and the motivations behind them. It's not quite a cut-and-dried "list of properties", but my impression is that the philosophies behind the various definitions are too complicated to just make a single list.