• A web site that has absolutely no inbound (backward) links to any of their pages.

This may make it more difficult to deal with the client because there will be a lot of work to build the inbound link authority. And, to intelligently do this will require some additional time from your client along with their patience.

• A web site that has been around for a bit but doesn't rank anywhere for their company name nor any valuable (searchable) keywords.

This one depends more on the reasons why the web site pages don't rank. It may simply be because the client neglected to actually talk about themselves somewhere on the site and/or is so into his product or service that he forgot to describe what it was. If this client doesn't understand that or suggests that the industry doesn't need explaining then it may be difficult to help get these needed optimization elements into his web site.

• A site that is completely designed in Flash.

This one says re-design. Is the client willing to incur possible additional expense and time for a redesign? And will he understand that if he allows SEO to work in tandem with the redesign, he'll be better off too.

If this project is all about the expense for this client, then he may cut corners on SEO.

• Signs that black-hat efforts have taken place on a site.
If the client doesn't know or won't admit to black-hat practices then he may not understand that white-hat practices may take time and are for the long haul. He may be used to the quick wins if he wasn't caught in time!