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Let's say your website sells MP3 songs. Your "browse music" page is key to helping visitors find what they want. The problem is that you have 15 possible column options. How do you display all the options so they are intuitive but not cluttered.
Ex options: (I made these up) Song Name, Artist Name, Producer Name, Years of producer experience, Producer age, Plays, Free Downloads, Purchases, Tempo, Genre, Category (separate from genre), Key (ex. c major), date added, comments, favorites.
Are we talking about sorting only or filtering too?
Sorting tends to be handled by clicking on the column header and, for simplicity's sake, only one column can be sorted on.
Your example sounds more like a case for applying filters. You might want to look at sites like eBay or LoveFilm for inspiration. They both offer tools for sifting through high volumes of data with multiple properties to filter by.
The core question is how would the user make a decision. To me it sounds like that sorting would not be a good solution in this case.
A lot of times, complexity comes from the fact that there is a mismatch between the user's mental model and the design. So user needs to learn how to use the interface which leads to greater cognitive load. (I should say that in some cases the extra cognitive load is worth it, if it adds value in other ways such as efficiency).
I would definitely encourage you to challenge this and define what problem you are solving first and then explore various solutions.
Personally I like the breadcrumb filter that NewEgg and other websites use when specifically filtering. That does not speak to how you should handle multiple sorts, like maintaining secondary sorts.
But for filtering, as your user makes a selection to filter results, add that selection to a breadcrumb list. If you want to get fancy, add a close type button (NewEgg uses an 'X') for each filter so that they can remove one if the results become too sparse.
Example below of a CPU search filtered by "Intel" then price.
