How do I use Assert to verify that an exception has been thrown?

For "Visual Studio Team Test" it appears you apply the ExpectedException attribute to the test's method.

Sample from the documentation here: A Unit Testing Walkthrough with Visual Studio Team Test

[TestMethod]
[ExpectedException(typeof(ArgumentException),
    "A userId of null was inappropriately allowed.")]
public void NullUserIdInConstructor()
{
   LogonInfo logonInfo = new LogonInfo(null, "P@ss0word");
}

Usually your testing framework will have an answer for this. But if it's not flexible enough, you can always do this:

try {
    somethingThatShouldThrowAnException();
    Assert.Fail(); // If it gets to this line, no exception was thrown
} catch (GoodException) { }

As @Jonas points out, this DOES NOT work for catching a base Exception:

try {
    somethingThatShouldThrowAnException();
    Assert.Fail(); // raises AssertionException
} catch (Exception) {
    // Catches the assertion exception, and the test passes
}

If you absolutely must catch Exception, you need to rethrow the Assert.Fail(). But really, this is a sign you shouldn't be hand-writing this; check your test framework for options, or see if you can throw a more meaningful exception to test for.

catch (AssertionException) { throw; }

You should be able to adapt this approach to whatever you like -- including specifying what kinds of exceptions to catch. If you only expect certain types, finish the catch blocks off with:

} catch (GoodException) {
} catch (Exception) {
    // not the right kind of exception
    Assert.Fail();
}

My preferred method for implementing this is to write a method called Throws, and use it just like any other Assert method. Unfortunately, .NET doesn't allow you to write a static extension method, so you can't use this method as if it actually belongs to the build in Assert class; just make another called MyAssert or something similar. The class looks like this:

using System;
using Microsoft.VisualStudio.TestTools.UnitTesting;

namespace YourProject.Tests
{
    public static class MyAssert
    {
        public static void Throws<T>( Action func ) where T : Exception
        {
            var exceptionThrown = false;
            try
            {
                func.Invoke();
            }
            catch ( T )
            {
                exceptionThrown = true;
            }

            if ( !exceptionThrown )
            {
                throw new AssertFailedException(
                    String.Format("An exception of type {0} was expected, but not thrown", typeof(T))
                    );
            }
        }
    }
}

That means that your unit test looks like this:

[TestMethod()]
public void ExceptionTest()
{
    String testStr = null;
    MyAssert.Throws<NullReferenceException>(() => testStr.ToUpper());
}

Which looks and behaves much more like the rest of your unit test syntaxes.