4. Moving silently is the ability to move with little sound and
        disturbance, even across a squeaky wooden floor, for
        instance. It is an ability which improves with xperience. <skill>

Moving Silently can be attempted each time the thief moves.

It can be
used to approach an area where some creature is expected, thus
increasing chances for surprise (q.v.). or to approach to back stab, or
simply done to pass some guard or watchman.
Failure (a dice score in excess of the adiusted base chance) means that movement was not silent
(see SURPRISE). Success means movement was silent. (SILENT MOVEMENT)

DMG:

Moving Silently: Silent movement is the same as normal exploratory movement,
    i.e. 12' per round as the thief creeps up (croodles) upon the AREA or victim or whatever.
Do NOT inform the thief that his or her dice score indicated a lack of success at this attempted stealth,
    if that is the case.
He or she thinks the movement is silent,
    and the monster or other victim will inform the character of his or her misapprehension soon enough.

Quote:
How did you handle situations where a character of one class (say Fighter) was attempting to use Move Silently or Climb Walls - Two situations that when taken literally (i.e. - the Thief can Move with ABSOLUTE SILENCE whereas someone else might attempt to move stealthily, and the Thief can climb SHEER SURFACES wheras someone else might climb a rough cliff, or a tree), would mean that only the Thief could attempt them, but when looked at in a broader fashion, might be allowed for a character of any class ?
 

Generally common sense was applied.
A fighter in metal armor can't move silently, but without that impediment a Dex ability roll with modifiers for surface and/or footwear, would be called for.