We flew when DS was 3 weeks, 3 months, 6 months, 13 months, and 27 months.
We did the lap-baby route for the first trip. The very first time, we checked the seat in up front. After that, we did the gate-check thing and I think we scored an empty seat.
From the second trip on, we have always bought him a ticket. I'm on the "if you can afford it, buy it" thought. With the recession, planes have been more fully booked, making the chances of scoring a free empty seat lower. It's been so bad that sometimes people have been bumped off of an overbooked flight, and unable to get rebooked for days.
I suppose I do agree with the FAA with regards to flying being safer than driving. So from that perspective, if you can't afford their own seat, by all means fly with a lap-child.
On the first trip or two, I used the ring sling during flight. It's not allowed for takeoff or landing, but it helped us with holding him and I hoped that if we did hit turbulence it would help keep him safe.
With an older child, I do my best to keep him in his carseat the full trip. Turbulence does happen, and injures people.