If my opponent plays Singularity on my Kai'Sa and she is at Star Spring and I react with a Nidalee with Ambush to summon her and then move Kai'Sa back to base with Star Spring, does the Singularity still kill my Kai'Sa?

Yes, Singularity will still kill your Kai'Sa.

According to Rule 359.3, Singularity does not have a location restriction—it simply says "Deal 6 to each of up to two units." Because the spell does not require the target to be at a specific location, moving the unit to your base does not make it an illegal target.

Here is the sequence of events:

1. Opponent plays Singularity: It is placed on the chain, targeting your Kai'Sa.
2. You play Nidalee (Ambush): You play Nidalee as a [Reaction] to a battlefield where you have units. This triggers Star Spring's ability.
3. Star Spring's ability resolves: You choose to move Kai'Sa to your base.
4. Chain resolves: The chain resolves in reverse order (LIFO).
   • Nidalee enters play.
   • Singularity resolves. It looks for its target, Kai'Sa. Since Singularity has no location restriction, Kai'Sa remains a legal target even though she is now in your base.
   • Singularity deals 6 damage to Kai'Sa, killing her.

This interpretation is supported by RiftJudge FAQ #3724, which clarifies that if a spell does not mention a location restriction, the unit remains a legal target even after moving to a different zone, such as your base.