If you choose to search your library but no applicable Aura card is left in your deck (or you choose not to find one), then nothing enters and you simply shuffle your library.

The attacking creature is not targeted by either this ability or the Aura, so a creature with shroud can be enchanted this way.

If the attacking creature has left the battlefield by the time Sovereigns of Lost Alara's second ability resolves, you can't put an Aura onto the battlefield this way. You may still choose to search your library just to shuffle it.

Any Aura card you find with Sovereigns of Lost Alara's second ability must be able to enchant the attacking creature as it currently exists. You need to check the Aura's enchant ability as well as any effects, such as protection, that would make it illegal to attach that Aura to the attacking creature. For example, if the attacking creature were an artifact creature with protection from blue, you could find an Aura with "enchant artifact," but you could not find a blue Aura.

Any Aura card you find with Sovereigns of Lost Alara's second ability must be able to enchant the attacking creature as it currently exists. You need to check the Aura's enchant ability as well as any effects, such as protection, that would make it illegal to attach that Aura to the attacking creature. For example, if the attacking creature were an artifact creature with protection from blue, you could find an Aura with "enchant artifact," but you could not find a blue Aura.

If you choose to search your library but no applicable Aura card is left in your library (or you choose not to find one), then nothing enters the battlefield and you simply shuffle your library.

If the attacking creature has left the battlefield by the time Sovereigns of Lost Alara's second ability resolves, you can't put an Aura onto the battlefield this way. You may still choose to search your library just to shuffle it.