3 sourced reports, with what to check before you buy. Each one links to the page it was reported on.
Costly or dangerous
AM04/AM05 printed-circuit-board short-circuit — Dyson fits a board protection part as the fix
Dyson Japan's official customer notice identifies the failing part as the printed circuit board, where an electrical short could cause product malfunction, and says the engineering fix is fitting a board protection component. This anchors the hazard to a concrete part.
What to check. The recall fix is a PCB protection part installed by Dyson technicians. A used AM04 that has not been through the recall program may still carry the unmodified board, so verify recall status before relying on a second-hand unit.
Dyson AM04 short-circuit recall — burned and melted internal parts
Dyson's own recall site confirms a voluntary worldwide recall of all AM04 and AM05 heaters because a small number short-circuited and produced burned/melted internal parts. This is the manufacturer's primary notice, so it is the strongest available evidence of a fire hazard in the AM04.
What to check. Before buying or using an AM04, check the serial number at dysonrecall.com. Affected units should be unplugged and sent in for the free engineering fix; Dyson says no injuries or property damage have occurred, but the short-circuit is a documented fire risk.
AM04 recall applies only to units bought before April 2014
The German-language recall page states that heaters bought after April 2014 should not be affected by the recall, giving EU buyers a practical cutoff date for the hazard.
What to check. For the EU market, a unit purchased after April 2014 is stated to be outside the recall; Dyson still asks owners to confirm by serial number at dysonrecall.com.
These reports come from the pages linked under each item: owner forums, repair guides and workshops. They are not a diagnosis of any one unit, and a careful check of the one you are buying still applies.
Text from the Kriko Electric space heaters catalog, version 0.1.1, under the CC-BY-SA-4.0 license.