My 1982 Peugeot PSVN has a bad front-end shimmy (wobble) when riding hands-off above roughly 15 mph. Doesn't matter if I am pedaling or coasting, and the pavement can be smooth. It stops if I put a hand on the bars. I've never noticed a shimmy at low speeds or if I have a hand on the bars.
Possibly related, the bike does an odd thing if I ride over a little bump (like a pavement ripple) when leaned over in a turn, this is with hands on the bars - the bike gives one little "shake of its head" then continues on its way. It isn't something I feel with my other bikes.
The bike is 58 cm, no fenders or other attachments. This is how it is equipped.
I am 5' 11", 186 lb. and usually have a messenger bag (it is my commuting bike).
What are some things I should check? So far all I've done is checked that the headset is not loose, if that even matters. I have searched here, but didn't find too much, other than to make sure fenders aren't loose - which is not applicable.