According to PHYSICAL CONSTANTS FOR INVESTIGATORS the ignition temperature of Cotton is 250°C at normal atmospheric pressure levels. Polyester has a much higher ignition temperature. If you happen to wear more exotic socks then please post letting me know what they are made of so I can complete my investigation, and with it, a risk assessment of your likelihood of survival.
Radiators will burn you long before they hit 80°C so I reckon I can without any fear of legal action advise that NO, your socks will not catch fire if you put them on a radiator. Campfires/BBQ’s/Gas Fires/Night Storage heaters/Bunsen Burners/Hobs/Lighters WILL burn your socks though so be extra careful when drying using any of the aforementioned heat sources.
HTH.
ALSO – Your kettle, whilst it provides heat is also WET on the inside so you can’t dry your socks in the Kettle. The fire risk is however, minimal.