Have a look at what is done commercially... Fig.1 illustrates a vintage 8255 TTL input/output board from the old IBM PC era - though very relevent to the type of DIL / TTL circuitry most PICAXE users are involved with. - Every integrated circuit on the board is decoupled with 100nF right up against its power input pins.
With regard to the type of decoupling capacitor see Fig.2. PICAXE boards appear to be supplied with 100nF polyester capacitors for decoupling.
Observing this, I usually have a 50+ stock of 100nF 63v 5mm Polyester box capacitors in my parts drawers (Part no. 10-3260 from Rapidonline.com) and I follow very much the advice of Inglewoodpete and put one on each and every power supply next to an active i.c.
PICAXE manual 3 advises using a 220nF polyester capacitor across the brushes of small electric motors. (Part no. 10-3264 from Rapidonline.com)
I have checked out the effect of the 220nF polyester capacitors with an oscilloscope on the the small motors that I commonly use and the effect is dramatic. Even so, after getting my fingers burnt with various forms of interference on the line from motors or actuators I now always design with two separate power supplies; one for the PICAXE and associated chips and a second one, usually from a secondary winding on the power transformer, for any motors or actuators. This seems to give 100% reliability.
I absolutely agree that the PICAXE technical reference manuals need a decent paragraph or two on the subject of decoupling. It is one of the most mentioned topics on this board and yet it is ignored in the PICAXE manuals. Obviously if only one PICAXE chip is being used in a simple battery-driven circuit and the inputs are switches, the outputs LEDs, then it is likely that the thing will work without decoupling. Add one interface chip to the story and then decoupling is a must.