Glad to hear you're making a custom board. This must be wildly exciting
Nothing pops out as being wrong. Can you link to the actual board you're using?
On the mfrc522 there is IRQ and NSS. I'm not sure if NSS or IRQ is the SPI chip select pin. Probably NSS (SS = slave select).
I assume the 3V3 and 3V3A on the spark core our outputs and don't need to be powered. Same for 3V3 and 3V32 on the VS1053.
All your ground and power pins look connected.
It looks like you're running two SPI busses on this device. If they are two hardware SPI busses, your good. If they're two software SPI busses, you're probably still good if speed isn't an issue. If there is only one hardware SPI bus the two devices should share MOSI, MISO, and SCK (SCLK is the same thing). Each one would need a unique chip select pin, which you already have.
Do you have your designs up on github? I'd love to follow what you do.