My notes on the issue, and I don't believe the situation has changed -
Summary
APA102 - Easily controlled directly from a PICAXE.
WS2801 - Directly controllable by a fast enough PICAXE with appropriate code.
WS2811 / WS2812 / WS2812B / APA104 - Not controllable by PICAXE.
APA102
These are the RGB LED strips we recommend for PICAXE use.
These devices use a two-wire SPI interface which can be easily driven from any PICAXE at any operating frequency. Hardware SPI (HSPIOUT), SHIFTOUT commands or bit-banged SPI can be used.
WS2801
These RGB LED strips can only be directly controlled by a fast enough PICAXE with appropriate code.
These devices use a two-wire SPI interface which requires sending bytes at a reasonably high speed. This is achievable with a PICAXE using hardware SPI (HSPIOUT) or internal SHIFTOUT commands.
However, the length of gap between consecutive bytes is used to reset the LED strip to indicate the next data is for the first RGB LED. A gap of greater than 500us is considered a reset. Only a PICAXE able to output consecutive bytes quickly enough is able to control a LED strip and the code has to be crafted to facilitate that.
PICAXE forum member 'westaust55' has kindly provided details and example code for using WS2801-based LED strips here -
www.picaxeforum.co.uk/showthread.php?22713-Getting-Started-with-the-WS2801
WS2811 /
WS2812 /
WS2812B /
APA104
These RGB LED strips cannot be directly controlled by a PICAXE.
The LED srips are similar to WS2801-based LED strips but have a shorter (50us) inter-byte reset period which is too short for the PICAXE to generate consecutive byte output without a reset occurring.