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Second, consistency of "story" throughout the process. Evidence suggests that users react best when the transition between steps of the process maintains a degree of consistency with respect to language and presentation. Users want the context of the overall action (logging into the SP from the IdP) to be clear at each step. If the UI changes in a jarring way, and/or lacks a visual and/or textual reference back to the overall goal/activity, they get confused and lost more easily. A number of new software features and federation initiatives (e.g., User Interface Elements) relate to this need, so we know improvement will be gradual.
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The discovery component may be part of the application or SP, or a separate service. The former enables a more consistent UI experience (color, style, etc.), while the latter enables work to be offloaded and shared by multiple services with identical or at least compatible discovery needs. In either case, the UI should remain as consistent with the originating application as possible. If the UI replaces that of the service, the UI should reference the service with some combination of descriptive name, icon, URL, etc. This provides the user with context for the selection of a login mechanism or IdP. When implemented separately from the service itself, SP user interface elements in federation metadata can supply the UI with this information.
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Other choices (i.e., the "rest") should be selectable primarily through a dynamic search text box, akin to most search engines. This scales well and is simple to use provided the user can enter a variety of terms, such as geography, organization names, mascots/brands, etc. If implemented consistently, one successful "find" is sufficient to teach a user how to find their choice somewhere else. When multiple results are found and displayed, tool tips containing more extensive descriptive text might be used to disambiguate like-named choices (but beware of mobile devices that don't support mouseover events). Icons may also be used in some cases. IdP user interface elements in federation metadata can help supply the UI with this information.
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The user interface of the IdP is not for the most part within the reach of standardization, but one can apply lessons learned to achieve a better, more consistent approach that will feel more natural to users. The most essential point is that the "story" of the login be maintained by identifying the service involved using a name, descriptive text, an icon, etc. SP user interface elements in federation metadata can supply the UI with this information.
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In keeping with earlier recommendations, a consent UI must identify the service receiving the information. SP user interface elements in federation metadata can supply the UI with this information, including links to privacy policies that are obviously relevant to the act of consent.
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The case of outright technical failure is discussed in the Error Handling topic. The Federated Error Handling topic discusses failure due to access control issues and the hand-off between IdPs and SPs.