Some bonsai can be very tastefully done in broken pots.
Most of the time however we want to avoid broken or breaking pots when purchasing our newest acquisition. Cracks and chips, aside from being unsightly, can lead to the complete failure of the pot further down the line.
Dirty pots can also be an indication of neglect, not only of the containers but also potentially of the trees.
Fine or end roots should not be visible, excessively exposed roots are not good for the trees overall health and can also indicate improper re-potting and or maintenance.
Containers should match the style of tree.
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Cascading bonsai should generally have deep upright pots to stabilize and counter the weight of the cascading tree. For other bonsai the pots are generally very shallow. The container should not distract or draw attention away from the bonsai and should be between two thirds and three quarters the height of the tree in length.
Proportions are very important in bonsai and need to be kept in mind. All aspect of the tree and its container need to be balanced and in proportion to enhance the overall effect.