How do you keep sequin appliques from puckering the base fabric?

You've diagnosed it right - the sequin fabric is basially a woven and the spandex moves, so any full-perimeter stitching locks them togather and the spandex fights back. I ran into the same wall doing metallic appliques on a dance costume base and the fix was two-part. First, yes, stretch the base while you stitch. Put the panel on the dress form and pin it out to roughly wearing tension - not drum-tight, just so the surface is taut and smooth. Then stitch each ray on while the base is held in that stretched state. That way the applique gets attached at the size the spandex actually is when it's on a body, instead of at its relaxed size, which is what causes the bunching once you take it off the form. If you've been stitching with the panel lying flat and relaxed, that's your ripple source right there.

Second, stop anchoring all the way around. Anchor at points - the tip and two or three spots along each side - with a small zigzag or a hand backstitch, and let the middle of the ray float. Between anchors the sequin piece can shift a hair while the spandex moves underneath, which is exactly what you want on curves. On the realy curvy areas I also eased in a tiny bit of fullness in the sequin piece itself, just a whisper, so it domes over the curve instead of bridging it flat. One more thing: trim the sequins back about a seam allowance's width from the cut edge of each ray so you're stitching through mesh only. Stitching through raised sequins forces the foot up, makes your tension go wierd, and adds bulk that reads as ripples.

Honestly, the full-panel interfacing prolly made things worse - it turned the whole base into a semi-woven, so now the stretch mismatch happens between the fused panel and the unfused rest of the dress. If your earlier rays look fine on the form but ripple off it, they may relax into place once the dress is on a body. I'd check the finished ones worn before unpicking anything.

Following this one closely since I'm doing beadwork on a stretch bodice right now and dealing with the same headache. One thing that's been saving me: I thread-trace where each motif goes while the base is on me, not the form, since my form's a size off and it threw my placement off the first time. Also worth checking if your thread has any give at all, a poly blend absorbs a little movement that cotton just refuses to.

Been through a version of this with lace motifs on a swimsuit base and the thing that finally stopped the ripples was switching to a walking foot. The regular foot was feeding the top faster than the spandex underneath and every anchor point came out with a tiny pucker, even with light pressure. With the walking foot both layers fed together and the ripple problem basically went away overnight. Might not solve the stretch mismatch on its own but it made my anchor stitches clean instead of adding their own bunching. Also I found hand stitching the curvy sections was easier than fighting the machine on those.

I made a stretchy dance top once and tried sewing stiff shiny shapes onto it while the fabric was just lying flat on the table, and the whole thing looked fine untill someone actually wore it, then everythign went wavy. Yours sounds way fancier than mine though. That bit about cutting the bumpy bits back from the edge before sewing explains a lot, my needle kept hopping over the raised parts and the stitches looked terrible. coud you skip sewing entirely and use fabric glue for the middle of each ray, then just stitch the edges? Or does glue crack when the stretchy part moves?

I don't buy the pin-it-to-the-form advice. I stretched a knit panel to wearing tension once to sew trim down, and the trim got locked in at the stretched length, so when the piece came off the form everything bagged and sagged instead of rippling. Swapped one problem for another. Spandex isn't taut evenly on a body anyway, it's snug in some spots and slack in others, so wich tension are you even matching? I had better luck stitching with the base totally relaxed and letting the trim absorb the movement, and fusing a scrap of stretch interfacing just under each applique rather than a whole panel. Fourteen rays is a lot to redo on a hunch, so I'd test one on scrap before committing either way.