How do I get the member to which my custom attribute was applied?

Attributes provide metadata and don't know anything about the thing (class, member, etc.) they are decorating. On the other hand, the thing being decorated can ask for the attributes it is decorated with.

If you must know the type of the thing being decorated you will need to explicitly pass it to your attribute in its constructor.

[AttributeUsage(AttributeTargets.Method | AttributeTargets.Property, 
    AllowMultiple = true)] 
public class MyCustomAttribute : Attribute
{
   Type type;

   public MyCustomAttribute(Type type)
   {
      this.type = type;
   }
}

Since there seems to be a lot of confusion with respect to how the stack frames and methods work, here is a simple demonstration:

static void Main(string[] args)
{
    MyClass c = new MyClass();
    c.Name = "MyTest";
    Console.ReadLine();
}

class MyClass
{
    private string name;

    void TestMethod()
    {
        StackTrace st = new StackTrace();
        StackFrame currentFrame = st.GetFrame(1);
        MethodBase method = currentFrame.GetMethod();
        Console.WriteLine(method.Name);
    }

    public string Name
    {
        get { return name; }
        set
        {
            TestMethod();
            name = value;
        }
    }
}

The output of this program will be:

set_Name

Properties in C# are a form of syntactic sugar. They compile down to getter and setter methods in the IL, and it's possible that some .NET languages might not even recognize them as properties - property resolution is done entirely by convention, there aren't really any rules in the IL spec.

Now, let's say for the moment that you had a really good reason for a program to want to examine its own stack (and there are precious few practical reasons to do so). Why in the world would you want it to behave differently for properties and methods?

The whole rationale behind attributes is that they are a kind of metadata. If you want a different behaviour, code it into the attribute. If an attribute can mean two different things depending on whether it's applied to a method or property - then you should have two attributes. Set the target on the first to AttributeTargets.Method and the second to AttributeTargets.Property. Simple.

But once again, walking your own stack to pick up some attributes from the calling method is dangerous at best. In a way, you are freezing your program's design, making it far more difficult for anybody to extend or refactor. This is not the way attributes are normally used. A more appropriate example, would be something like a validation attribute:

public class Customer
{
    [Required]
    public string Name { get; set; }
}

Then your validator code, which knows nothing about the actual entity being passed in, can do this:

public void Validate(object o)
{
    Type t = o.GetType();
    foreach (var prop in
        t.GetProperties(BindingFlags.Instance | BindingFlags.Public))
    {
        if (Attribute.IsDefined(prop, typeof(RequiredAttribute)))
        {
            object value = prop.GetValue(o, null);
            if (value == null)
                throw new RequiredFieldException(prop.Name);
        }
    }
}

In other words, you're examining the attributes of an instance that was given to you but which you don't necessarily know anything about the type of. XML attributes, Data Contract attributes, even Attribute attributes - almost all attributes in the .NET Framework are used this way, to implement some functionality that is dynamic with respect to the type of an instance but not with respect to the state of the program or what happens to be on the stack. It is very unlikely that you are actually in control of this at the point where you create the stack trace.

So I'm going to recommend again that you don't use the stack-walking approach unless you have an extremely good reason to do so which you haven't told us about yet. Otherwise you are likely to find yourself in a world of hurt.

If you absolutely must (don't say we didn't warn you), then use two attributes, one that can apply to methods and one that can apply to properties. I think you'll find that to be much easier to work with than a single super-attribute.

GetMethod will always return you the function name. If it is a property, you will get either get_PropertyName or set_PropertyName.

A property is basically a type of method, so when you implement a property, the compiler creates two separate functions in the resulting MSIL, a get_ and a a set_ methods. This is why in the stack trace you receive these names.

custom attributes are activated by some code calling the GetCustomAttributes method on the ICustomAttributeProvider (reflection object) that represents the location where the attribute is applied. So in the case of a property, some code would obtain the PropertyInfo for the property and then call GetCustomAttributes on that.

If you want to build out some validation framework you would need to write the code that inspects types & members for custom attributes. You could for example have an interface that attributes implement to participate in your validation framework. Could be as simple as the following:

public interface ICustomValidationAttribute
{
    void Attach(ICustomAttributeProvider foundOn);
}

Your code could look for this inteface on (for example) a Type:

var validators = type.GetCustomAttributes(typeof(ICustomValidationAttribute), true);
foreach (ICustomValidationAttribute validator in validators)
{
     validator.Attach(type);
}

(presumably you would walk the whole reflection graph and do this for each ICustomAttributeProvider). For an example of a similar approach in action in the .net FX you can look at WCF's 'behaviors' (IServiceBehavior, IOperationBehavior, etc).

Update: the .net FX does have a sort-of general purpose, but basically undocumented interception framework in the form of ContextBoundObject and ContextAttribute. You can search the web for some examples of using it for AOP.