General:
* Shred
* Rag
* Fragment
* Piece
* Scrap
* Remnant
* Bit
More specific:
* For clothing: Torn, tattered, frayed, ragged, worn, threadbare
* For things in general: Broken, shattered, ruined, dilapidated, derelict
The best synonym will depend on the context of the sentence. For example, if you're talking about a piece of torn fabric, "shred" would be a good synonym. If you're talking about a building in a state of disrepair, "dilapidated" would be a better choice.