Is an AC transformer in a desk light from the 1970s a significant risk to keep using in your home when it comes to fire safety? It was already broken in the first place so this is a theoretical question. When it was still working a few years back, I could hear the mains hum coming from it. My annoyance with that caused me to neglect the lamp. Picture of the transformer in question below:
If it's properly designed (the original lamp-transformer system) it won't be much of a fire hazard or shock hazard. A 1970s lamp probably is made of metal, so even if the transformer failed it would be unlikely to be able to start a fire- the winding or a fuse (internal or external to the transformer) will fail first.
The transformer you show is of split-bobbin design which provides a lot of insulator between primary and secondary windings (at the expense of some performance).
A hum or buzz from the transformer (typically at 100 or 120Hz depending on your mains frequency) isn't typically a sign of impending doom, it's just a loose lamination or something similar moving from the magnetic field, which does not noticeably affect the transformer operation.
The biggest indications of safety are UL/CSA or similar markings, and the cord and housing being in good condition. Also, if the bulb is a halogen type, the glass envelope has to get to a very high temperature compared to conventional incandescent lamps (>500°C) in order for the cycle to work- hot enough to perhaps ignite something. As we all may expect from Ray Bradbury's classic dystopian novel title, paper can ignite at 233°C. There should be protective glass preventing any contact with potentially flammable materials even if the lamp is knocked over by a pet or whatever. That's a safety requirement that may not have existed when the lamp was manufactured, and quite a few fires happened before it was codified if memory serves.
Once you remove the transformer from the product it could be more dangerous since a failure might not be contained by a metal housing. Or maybe you could drop it on your foot. Or the mains wiring might not be properly insulated or fused. Lots of ways to make it unsafe.
In any case, nothing is 100% safe. Inside an unmodified and relatively modern (1970s) safety-agency approved product it should be relatively safe.
As Spehro says - the transformer is likely to be safe.
However: (and not what you asked about)
The lamp itself may be a fire hazard.
They say:
There are no wires between the base and the lamp: the two parallel arms, connected with snap fasteners, conduct safe 12-volt electricity
The use of "snap fasteners" suggests that the current is transferred across use disconnectable connectors. There are numerous lamp designs which transfer energy along bare conductove metal arms or rods. The 12 VAC can be directly connected to. This is a potential but usually unrealised fire hazard. I have tried to produce fire from such lamps to see how likely this is in use. I could do so only with significant effort. So, in practice this aspect also seems "reasonably safe" - but still a hazard.
Here is a relevant paper.
Note that the mechanisms that allow very low ignition energy will often be absent in lamp use environments. But, not always.
Low Voltage: The Incompetent Ignition Source - Dispelling the Myth
Old transformers can contain poly-chlorinated biphenyls (PCBs, not to be confused with Printed Circuit Boards) which are highly carcinogenic.
However this is more relevant to utility-size transformers which have liquid cooling, rather than small transformers used in domestic appliances. But if the transformer has any kind of oil in it (even sealed inside) then there is a risk the oil may contain PCBs.
All electrical transformers are AC. That's how they work. The buzzing should be harmless, unless there is a short circuit somewhere. But all non potted transformers produce some sound, much in the way a loudspeaker does. It's the interaction of the current flowing through the coils causing some minor movement in the coil an/or the laminations. Your local electricity substation also hums, when it's working perfectly well, if you listen carefully.
But to make it much safer, put that transformer into a suitable enclosure, preferably with ventilation, so that nobody can touch any high voltage terminals or wiring. It's also good practice to add a suitable fuse to protect from sustained overload currents.
UL certifies the entire fixture as safe, including the use of that transformer in that manner in that enclosure with that wiring.
If you salvaged the transformer and are placing it to another use, all bets are off... unless you send several specimens with a check for $20,000 to UL to have your design UL Listed.
Or you could simply follow the UL standards (e.g. UL White Book) in your design choices, and feel assured that it is just as safe as if UL actually checked your work. (true or not lol).
Depending on material condition, of course. If you had a transformer whose health you were unsure of, you could certainly use a mega-ohmmeter to test (an ohmmeter with a test probe voltage of several hundred volts), and I suppose if it passes you could send it out to a friendly neighborhood motor shop for VPI treatment - that's placing the unit in a vat of epoxy, drawing vacuum followed by pressure to force the epoxy into all voids. That's good to reduce vibration. Or feel free to replace the transformer with a UR-Recognized electronic power supply, in a manner which UL would approve.
