Let's look at the common ones first, *nonperishable* and *imperishable*:
> **nonperishable**: not subject to rapid deterioration or decay. [source][1]
>
> *A supply of nonperishable food was kept for emergencies.* 

*Nonperishable* and its typically plural noun form, *nonperishables*, are used for concrete objects, like food, supplies, and medications, that are intended to last a long time.

>   **imperishable**: not subject to decay; indestructible; enduring. [source][2]
>
>  *Though Thelonius Monk died decades ago, his imperishable legacy arouses our senses still today.* 

I mostly see *imperishable* used in a literary or abstract context. It can also be used for items, but this usage is less common.

*Unperishable* means the same as *nonperishable*, but is the least common of all ([source][3]). I've never seen it used.

[![enter image description here][4]][4]


  [1]: http://www.dictionary.com/browse/nonperishable
  [2]: http://www.dictionary.com/browse/imperishable?s=t
  [3]: https://books.google.com/ngrams/graph?content=non%20-%20perishable%2Cunperishable%2Cnonperishable&case_insensitive=on&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t4%3B%2Cnon%20--%20perishable%3B%2Cc0%3B%2Cs0%3B%3Bnon%20--%20perishable%3B%2Cc0%3B%3BNon%20--%20perishable%3B%2Cc0%3B.t1%3B%2Cunperishable%3B%2Cc0%3B.t4%3B%2Cnonperishable%3B%2Cc0%3B%2Cs0%3B%3Bnonperishable%3B%2Cc0%3B%3BNonperishable%3B%2Cc0
  [4]: https://i.sstatic.net/dITHY.png