Do I get Windsinger's effect if I play it to base?

Yes, you do get Windsinger's "When you play me" effect if you play it to your base.

Windsinger's effect is a triggered ability: "When you play me, you may return another unit at a battlefield with 3 [S] or less to its owner's hand."

Here is the breakdown of why this works:

1. Playing the Unit: When you play Windsinger to your base, it enters the board.
2. Triggered Ability: Once Windsinger has entered play, its triggered ability is placed on the chain.
3. Resolution: You resolve the ability. Because the ability allows you to target "another unit at a battlefield," you can select any legal unit currently located at any battlefield, regardless of where the Windsinger itself was played.

Important distinction: This is different from playing a unit from a Hidden state, where specific rules (such as FAQ #10647) may restrict targeting to the battlefield where the unit was hidden. When you simply play Windsinger normally from your hand to your base, you are not restricted by those hidden-zone rules, and you may target units at any battlefield.