When to use "me" or "myself"?

Which one is correct:

Someone like me...

or

Someone like myself...

Is "like myself" ever correct?

"Someone like me" is the correct one.

There's a lot to say about the usage I guess, but to make a long story short:

Me is a so-called objective pronoun, opposed to subjective pronouns (I, you, she, he). It's called like this because it's placed in the object field after verbs or prepositions:

Wait for me!
She likes me.

Myself instead, is used with reflexive verbs, meaning those verbs that indicate an action that "falls" on the subject:

I wash myself.
I told myself it couldn't be true.

There are exceptions, you can find them here, but I'll paste the interesting part:

Usage note: There is no disagreement over the use of myself and other -self forms when they are used intensively "I myself cannot agree" or reflexively "He introduced himself proudly". Questions are raised, however, when the -self forms are used instead of the personal pronouns ( I, me, etc.) as subjects, objects, or complements. Myself occurs only rarely as a single subject in place of I: Myself was the one who called. The recorded instances of such use are mainly poetic or literary. It is also uncommon as a simple object in place of me: Since the letter was addressed to myself, I opened it. As part of a compound subject, object, or complement, myself and to a lesser extent the other -self forms are common in informal speech and personal writing, somewhat less common in more formal speech and writing: The manager and myself completed the arrangements.
There is ample precedent, going as far back as Chaucer and running through the whole range of British and American literature and other serious formal writing, for all these uses. Many usage guides, however, state that to use myself in any construction in which I or me could be used instead (as My daughter and myself play the flute instead of My daughter and I) is characteristic only of informal speech and that such use ought not to occur in writing.

My observation would be that Someone like me is fairly "neutral" or unemphatic, whereas Someone like myself is a more emphatic, almost as though you're "pointing" to yourself. (A bit like when you say "Myself, I think that...".)

It is never correct to use 'myself' in a sentence that does not also contain another first-person pronoun such as 'I' 'me', or 'my'. Acceptable examples are the reflexive use: "I did this myself." and the intensive case: "I myself did this." Here's a great article on the subject. EDIT: My original rule was incomplete - see comments below.