If an effect says "You may search your library . . . If you do, shuffle your library," you can't choose to search since it's impossible, and you won't shuffle.

If an effect says "Search your library . . . Then shuffle your library," the search effect fails, but you will have to shuffle.

Since players can't search, players won't be able to find any cards in a library. The effect applies to all players and all libraries. If a spell or ability's effect has other parts that don't depend on searching for or finding cards, they will still work normally.

Effects that tell a player to reveal cards from a library or look at cards from the top of a library will still work. Only effects that use the word "search" will fail.