[c#] What Makes a Method Thread-safe? What are the rules?

If a method (instance or static) only references variables scoped within that method then it is thread safe because each thread has its own stack:

In this instance, multiple threads could call ThreadSafeMethod concurrently without issue.

public class Thing
{
    public int ThreadSafeMethod(string parameter1)
    {
        int number; // each thread will have its own variable for number.
        number = parameter1.Length;
        return number;
    }
}

This is also true if the method calls other class method which only reference locally scoped variables:

public class Thing
{
    public int ThreadSafeMethod(string parameter1)
    {
        int number;
        number = this.GetLength(parameter1);
        return number;
    }

    private int GetLength(string value)
    {
        int length = value.Length;
        return length;
    }
}

If a method accesses any (object state) properties or fields (instance or static) then you need to use locks to ensure that the values are not modified by a different thread.

public class Thing
{
    private string someValue; // all threads will read and write to this same field value

    public int NonThreadSafeMethod(string parameter1)
    {
        this.someValue = parameter1;

        int number;

        // Since access to someValue is not synchronised by the class, a separate thread
        // could have changed its value between this thread setting its value at the start 
        // of the method and this line reading its value.
        number = this.someValue.Length;
        return number;
    }
}

You should be aware that any parameters passed in to the method which are not either a struct or immutable could be mutated by another thread outside the scope of the method.

To ensure proper concurrency you need to use locking.

for further information see lock statement C# reference and ReadWriterLockSlim.

lock is mostly useful for providing one at a time functionality,
ReadWriterLockSlim is useful if you need multiple readers and single writers.