Completely ignoring any temptation to guess at the specifics of what you're trying to accomplish with your controls, or why, let me take a quick stab at answering your basic questions.
Up front; from your description, I think it sounds like your best option is just to have an instance field, not a static one, and not a singleton. You said "I actually need each instance of the control to have one such variable, not one variable across all the instances of the control in that single form." That is kind of a loose definition of an instance variable. Does it even really need to be public?
Actually, I would make it an instance property, because chances are pretty good that sooner or later you'll want to raise some event when the thing changes, or do something else that will end up making a public field feel like an unfortunate decision in retrospect. Hindsight is always 20/20. Or at least 20/30 or so. But in this case, a little foresight says that using a simple automatic property is cheap insurance:
public SomeDataType WidgetSettings{ get; set; }
So; there is only one instance of a static field/variable for the entire assembly. Every instance of the class containing that public static variable has a reference to the very same object in memory. So yes, all of your control instances are accessing the very same object/memory.
Any time you see "static" and "public" together in the same variable declaration line, it should be reason to pause and think. The thing is, if the variable is something like a string, or a numeric type, or a simple collection, you will have to use synchronization code everyplace that you access that field from. On the other hand, if it's a reference to a class with static getters and setters for various attributes, then you can put that synchronization code in the getters and setters, and you can test that class and its attributes in isolation (as a unit) and know that it works properly. I don't know if I expressed that as well as I meant to... But honestly, if you can what you need to do using instance variables and avoiding the whole thread-safety issue altogether, then do that instead.
If you have a static variable, it is absolutely for certain not thread-safe. At all. Maybe you're thinking that you're doing a Winforms app and all of the UI stuff executes on a single thread (because Winforms is not thread-safe), which is true enough if you follow the rules, except that we're all creative about using timers and background workers and such to loosen things up. So it is still possible for strange things to happen. So in the worst case, not only are all of your control instances accessing and updating the same variable, they might be trampling all over each other in the process.
If you're messing with WPF, then the UI library is thread-safe and the potential pitfalls of multithreading combined with static fields apply.
If the data that the variable points at is a type that requires more than one instruction to write completely, then it is only a matter of time before one control instance writes only half of a new value before the next control instance reads that field and gets a value composed of half of one write and half of a different write. I think the proper technical term for that situation is something along the lines of "fubar." Of course the thread that reads that mangled variable may well try to write back to it before it gets swapped out by the thread scheduler and the original thread gets another go, replacing the first thread's half-value with its own half-value before the first thread even gets a chance to write the second half of its value. This is likely to end in tears, or baldness, or more likely both tears and baldness.
Finally, the singleton has all the same potential thread-safety issues as the static field, because the whole notion of a singleton is that you grab your instance from a factory method, which serves the same single instance up to everybody who asks. So everybody who asks is, by definition, accessing the very same object.
In this case, my gut tells me that "singleton" is a code word for "fancy static field, which requires a lot more code and a lot more testing, and still isn't thread-safe unless you write error-prone and performance-robbing synchronization code to go with it."
When a code word has a definition that long, it's worth seriously looking at a different approach.
For what it's worth. :-) Without more detail it would be awfully difficult to provide any answers that relate directly to your question. What type of object is this static field pointing at? Is it a collection? Is it some kind of settings or configuration class with multiple properties? Is it an object that communicates with other parts of the system (events) in response to different signals you send it? Etc.
There are a few different approaches you can use to do this, depending on the data set, performance needs, and your general preference.
If there's an object that is used almost exclusively for display, you would probably want to query the dataset beforehand in some sort of Factory and then create the object by passing the data into the constructor.
If there's an object that touches a bunch of data sources and is used for different things, loading all of the data at instantiation might take a long time, and all of the object's data might not be needed every time that object is instantiated. You can leverage lazy loading to accomplish this goal.
If performance is a major concern, and the object stores a ton of data but is mostly immutable, you can use something like the Flyweight pattern, which allows you to actually treat a single object like a collection.
I highly recommend you try a few different ways of doing things. For a project such as yours, how you choose to instantiate and populate the object will probably matter little, but it will be a good and fun exercise in different object instantiation patterns.
As a final note, much of the knowledge you're gaining now about OOP can be applied to nearly any language you learn in the future.
Best Answer
Your question is a little vague so I'll try my best.
There's one good question to ask when you're trying to choose between creating a static or an instance method.
Static methods are no different than traditional procedural functions. You give it an input and it should consistently product an output based on the input that is completely side-effect free.
Basically, you could use a traditional function declaration in it's place with no difference. The only benefit is that static methods are attached to a namespace where traditional functions are defined in the global namespace.
They don't matter so much in PHP because the language mixes both procedural and OOP styles but in languages that are pure OOP (ex C#/Java) there is no way to define functions so static methods are used in their place.
Static methods are perfectly acceptable to use when you're defining helper methods that act in a stateless manner. If you need data persistence between calls and/or find yourself relying on global variables, don't; use objects/instances instead.
Update:
Tom B made a very good point about unit testing. That static methods (like procedural functions) are inherently decoupled from the classes that they're attached to. So, if a static method fails, it may also cause any other instance method it's called in to also fail. It's a classic example of a side-effect.
That explains why I have encountered so many people parroting 'static methods are bad, mmmkay' in the TDD community. They have a completely valid point.
There are ways this could be improved. In the language, throwing an implicit exception when static methods fail. Technically, it wouldn't be any messier than doing type conversions. Assume it's going to break something if it fails. Of course, that assumes a higher degree of understanding to implement.
Either way, it comes down to the simple question.
My personal bias is toward expressiveness, the TDD school of thought's bias is toward consistency. YMMV.
I look forward to many years of religious wars over this topic.