Does Lapse of Certainty put the targeted spell on top of its owner's library immediately or at end of turn?

Lapse of Certainty reads: "Counter target spell. If that spell is countered this way, put it on top of its owner's library instead of into that player's graveyard." Both effects happen at the same time as part of the spell's resolution — there is no delayed trigger or end-of-turn clause.

When a spell is countered, it would normally move to its owner's graveyard as a state-based-like result of being countered (CR 608.2b, 700.4). Lapse of Certainty's replacement effect intercepts that move and redirects the card to the top of the library instead, all during the resolution of Lapse of Certainty itself.

This is governed by CR 700.4, which explains that when a spell is countered, it goes to the graveyard unless another effect specifies a different zone. Lapse of Certainty provides exactly that alternative destination, and it takes effect immediately upon resolution.

Example: Your opponent casts Wrath of God. You cast Lapse of Certainty targeting it. Lapse resolves: Wrath of God is countered and placed on top of your opponent's library right now — not at the end of the turn. Your opponent will draw it again next turn if nothing else intervenes.

There is no timing trick here; the card goes to the library the moment Lapse of Certainty finishes resolving (CR 608.2b). Players should be careful: your opponent will have access to that card again as soon as they draw next.