Can you cycle a card while another spell is on the stack?

Cycling is an activated ability, not a spell. It follows the rules for activated abilities, not for casting spells. Because the reminder text says you may cycle at any time you could play an instant, you can activate it while another spell or ability is already on the stack — even during your opponent's turn or in the middle of combat.

When you cycle, you pay the cycling cost and discard the card as the cost. The cycling ability then goes on the stack on top of whatever is already there. Both players get priority in the normal order before it resolves. When it resolves, you draw one card (or trigger an additional effect if the card has a triggered cycling ability).

This is governed by CR 702.29 (Cycling) and CR 509.2 (Activated abilities can be activated any time the player has priority, with timing restrictions based on whether the ability has 'activate only as a sorcery' language). Since cycling carries no such restriction, it follows instant timing.

Concrete example: Your opponent casts Doom Blade targeting your creature. In response, before it resolves, you cycle Irrigated Farmland by paying its cycling cost. The cycling ability goes on the stack above Doom Blade, resolves first, and you draw a card — all before Doom Blade resolves.

Note: If a card with cycling also has a triggered ability that triggers when you cycle it (e.g., 'When you cycle this card, you may...'), that trigger also goes on the stack at the same time and follows the same rules.