There are a couple of issues with Singletons. I'll outline them below, and then propose some alternative solutions. I am not a "Never use a singleton, or you're a crap coder" kind of guy as I believe they do have their uses. But those uses are rare.
Thread safety. If you have a global-static Singleton, then it has to be thread-safe because anything can access it at any time. This is additional overhead.
Unit Testing is more difficult with Singletons.
It's a cheap replacement for global variables (I mean, that's what a singleton is at the end of the day, although it may have methods and other fancy things).
See, it's not that Singleton's are "horrid abominations" per-se, but that it's the first design pattern many new programmers get to deal with and its' convenience obfuscates its' pitfalls (Just stick some gears on it and call it steam-punk).
In your case, you're referring to Models and these are always "instances" as they naturally reflect the data. Perhaps you are worried about the cost of obtaining these instances. Believe me, they should be negligible (down to data-access design, obviously).
So, alternatives? Pass the Model to the places that require it. This makes unit testing easier, and allows you to swap out the fundamentals of that model in a heart-beat. This also means you might want to have a look at interfaces - these denote a contract. You can then create concrete objects that implement these interfaces and voila - you're code is easily unit-testable, and modifiable.
In the singleton world, a single change to that singleton could fundamentally break everything in the code-base. Not a good thing.