Does countering a spell with Counterspell prevent its triggered abilities from going on the stack?

When a spell is countered, it is moved from the stack to its owner's graveyard without resolving (CR 701.5a). Because the spell never resolves, any abilities that are written as 'when this spell resolves' simply never trigger — there is no resolution event to cause them to trigger.

However, it is important to distinguish triggered abilities that trigger on casting from those that trigger on resolution. Abilities that say 'when you cast this spell' trigger at the time the spell is cast and go on the stack immediately (CR 603.6a). Those triggers have already gone on the stack before Counterspell is even cast, so countering the original spell does NOT prevent those 'when cast' triggers from resolving.

Additionally, some spells have triggered abilities that are part of the spell itself but only function once the permanent enters the battlefield. Since a countered spell never becomes a permanent, those enter-the-battlefield triggers never occur either (CR 603.3).

Concrete example: Your opponent casts a creature with 'When you cast this spell, draw a card.' You respond with Counterspell. The 'when cast' trigger has already gone on the stack, so your opponent still draws a card. But if the creature had 'When this spell resolves, create a token,' that ability never triggers because the spell never resolves.

In summary: countering a spell stops resolution-based triggers and ETB triggers, but cannot retroactively remove 'when you cast' triggers that have already been placed on the stack.