The bottom basin and what I used to drain out all of its mineral oil.
QUICK METHOD. To dump the oil out of the bottom basin quickly without cleaning that basin, it should be possible to simply tilt the lamp sideways towards some receptacle, hopefully with the lamp unplugged, switch out of the way, and foliage removed. The oil should easily come out of the larger holes of the tray, enough to pour from the lip of the bottom basin. But I suspect that method could be messy.
SLOW METHOD. If time permits and cleaning is desired, I would recommend first removing the lower basin entirely by unscrewing the lower basin from the cage part of the lamp. First prepare a location to place the dripping cage bottom as it comes off, such as on top of a large trash can liner laid on the floor with newspapers on top. Be careful not to let the switch come into contact with any mineral oil. If time permits, wrapping plastic around the switch with absorbant paper towels underneath might be a good idea, too.
Applying the slow method, a method of draining that worked for me was to use the oil funnel for my car oil and a discarded plastic milk jug, all set atop some newspapers, and to pour the basin's oil into the milk jug via the funnel. I then used dry paper towels to get the worst sludge off the basin (it's like mud!), followed by dishwashing liquid and wet paper towels, followed by cleanser and wet paper towels.
The interior of my bottom basin was permanently stained by decades of dirty oil, but it doesn't really matter since the basin interior is ordinarily entirely hidden from view. I suspect that cleaning the basin interior will affect the cleanliness of the rest of the lamp, though, since the main source of dirt in the lamp is dirty oil being distributed to all parts of the lamp.
The above shaking method could be done just as well with a sealed Tupperware container, but the thought of putting such dirty old mineral oil into food dispensers didn't appeal to me. Letting the foliage soak in a sink or tub might be alright, too, except that tiny plastic leaf fragments could get lost and/or get lodged in a drain or garbage disposal.
Although the underside of the lid is not normally seen, it can be smelled. The entire top of the lamp can have an odor of rancid mineral oil. I haven't yet been able to completely eradicate the odor, although I believe using the brass cleaner helped.
Then later, with the lamp turned off, you can then just stick a sewing needle down the holes of the marked spouts, even with the fiber inside. Just rub the needle around inside the spout in a circular motion.
The only trouble I had is that I accidentally dropped the needle once while cleaning a spout this way while the lamp was running, and the needle quickly slid through the spout and all the way down to the bottom of the strand, though fortunately it lodged against the pedestal and didn't go through the bottom hole into the bottom basin. It is more dangerous to clear spouts this way while the lamp is running, but the advantage is it will be immediately apparent if the hole was sufficiently cleared enough by observing the ensuing drip pattern. I'd recommend wearing rubber gloves and being sure that the socket connections at the wire nuts are well insulated with electrical tape and not dipping into the oil or touching any metal, to avoid the risk of electrocution.
Another trick I discovered is to shine a light up at the bottom of the top basin so that the light can be clearly seen through the spouts when viewed from above. This can be done easily when the lamp is running, too, though be careful not to let the light source you're using touch the dripping oil on the strands. Any dark holes will then be readily apparent, and it might not even be necessary to mark them before unclogging them with a needle. This also creates sort of a cool psychedelic effect, with the bright points of light blinking off and on as the droplets form and fall away.
Immediately after cleaning my rain lamp, I found it useful to initially test the lamp by running it with the bulb removed and lid unscrewed. That way any spouts that are clogged can be easily marked by putting tape alongside them, then later the clogged spouts can be quickly identified and cleaned when the lamp is unplugged, without having to go through the messy bulb removal process and lid unscrewing process. I also found that by watching the top basin while the lamp is running, one can see small bits of debris like metal shavings and tiny dust balls floating in the oil that weren't present or visible before running. Such debris can then be removed by dipping a paper towel into the oil to pull them out so that they don't later clog any spouts.
Paper towels, facial tissue, or toilet paper all work fine for the remaining areas, especially if used with hot water and soap. I was afraid to use anything other than cold water on the statue, though, for fear of damaging the original paint job. I didn't need to remove the fiber for any of the cleaning, although I suppose if it were removed (not worth the trouble!), cleaning would be faster and better. I wiped some dirty oil off the surfaces of the pump and motor, but didn't attempt any special cleaning of the motor. I wouldn't want to risk rusting any metal parts of the lamp by putting parts in water. However, one Old-House-Friends Forum member reports successfully cleaning the entire lamp (except the motor) that way. Other tips from the Old-House-Friends Forum: