Churchill's funeral barge on the Thames, with the cranes being dipped out of respect. That's the very first thing I ever remember seeing on TV (on a 10" or 12" B&W English Electric valve set I think).
On a much lighter note, the baby elephant crapping and urinating live on air all over the studio floor on 'Blue Peter', with the keeper slipping on the sludge as he tried without success to haul the beast in, was pretty unforgettable. Children's TV at its best !