How safe are mobile phone batteries?

Episode 1330 (39:30)

Graham from London, England
Samsung Battery

Graham is worried about the lithium-ion batteries that phones use. Leo says that everything uses lithium-ion these days. In the Tesla, they have protected the car by putting titanium barriers between the lithium-ion batteries and the rest of the car. They are dangerous if punctured because they will instantly catch fire.

(Image By Dmitry G (Own work) [CC BY-SA 3.0 or GFDL], via Wikimedia Commons)