When I applied for my first Amex, I got a great bonus applying online.
When I applied for my second, I used some funny link not meant for existing customers and somehow didn't get the bonus ($100)
. I've milked it for some good offers, though, while hardly giving Amex any real business on the card.
When I applied for my third, I applied over the phone and asked the CSR to check exactly what bonus I was eligible for. I got it without any trouble.
I suppose if you have mixed/irregular/unusual sources of income, then applying over the phone might be helpful since a (smart) CSR can help you determine what to include. If the CSR doesn't sound smart, I'd apply online.