C# – Is it possible to write a regex that does one search then uses its results to do another search

Given your code, I have two concerns.

First, readonly references to mutable child objects (like collections) do not make the properties of the child immutable.

public class Foo
{
   public readonly List<int> Bar = new List<int>{1,2,3};
} 

...

var foo = new Foo();

//You may enumerate Bar and read its indices
foreach(int baz in foo.Bar) Console.WriteLine(baz.ToString());

var bob = foo.Bar[1];

//this is illegal because the reference is readonly
foo.Bar = new List<int>{4,5,6};

//but the properties of the immutable reference are still mutable, so this is fine:
foo.Bar.Clear();
foo.Bar.Add(4);
foo.Bar.Add(5);
foo.Bar.Add(6);

If you want a truly immutable child object, you have to make the reference immutable AND make sure the object itself is immutable. For that, you typically want to create or use a read-only version of your child and return that instead:

public class Foo
{
   private readonly List<int> bar = new List<int>{1,2,3};

   public ReadOnlyCollection<int> Bar { get { return bar.AsReadOnly(); } }
} 

...

var foo = new Foo();

//You may still enumerate and index Bar
foreach(int baz in foo.Bar) Console.WriteLine(baz.ToString());

var bob = foo.Bar[1];

//this is illegal because there's no setter
foo.Bar = new List<int>{4,5,6};

//And these are also illegal because ReadOnlyCollection doesn't expose any such methods:
foo.Bar.Clear();
foo.Bar.Add(4);
foo.Bar.Add(5);
foo.Bar.Add(6);

Second, the locked, double-checked instantiation of the singleton is (much) better than nothing for making it thread-safe, but there are still some cases where this can fail to prevent double-instantiation. Microsoft instead recommends use of a static readonly field with instantiation defined:

public sealed class MySingl
{
   public static readonly MySingl Instance = new MySingl();

   private MySingl() { ... }
}

Microsoft guarantees an implementation like this to be thread-safe in the Microsoft CLR; statics are instantiated just-in-time and the CLR has internal mechanisms to block other threads that need this reference while it's being created.

Data is now (after parsing) stored in just a list, searching for a specific peace of data is not very well optimized. Are there other structures I could use, which I can search using file(string) - level(integer) - sub-level(integer) so I can quickly get a specific data object.

Assuming that you want to search by File, Level, Sub-Level only, you have a clear hierarchical structure in that description, it sounds like you can divide one ArrayList into multiple steps. You could even make each hierarchy level a class itself to make it more clear, for example:

class SomeFileData {
   List<LevelData> levels;
}

class LevelData {
   List<SubLevelData> sublevels;
}

class SubLevelData {
   // Probably similar to your current `FileData` implementation
}

And if you need to save multiple files at once, you can use a Map with the file name as key.

class LotsOfFileData {
    Map<String, SomeFileData> files; 
}

Using this hierarchy, you can easily get file "mydata.dat", level 3, sublevel 4.

LotsOfFileData allData;
allData.getFile("mydata.dat").getLevel(3).getSublevel(4)

Each step in the get process looks at the current class' list or map to retrieve the data.

By the way, declare variables according to their interface, not their implementation. Declare your ArrayLists as List to allow for easy change of implementation.