If it's a serious oil/grease cleanup, then
Odorless Mineral Spirits in a
lab type squirt bottle.
Add an old toothbrush to cut through it, if it's really caked on grunge.
Bend the head somewhat backward with a heat gun, for better reaching.
Bottle brush if it's in crevices.
Add a compressed air gun, if you really want it spotless NOW, or in deep recesses. (Shield you eyes, trust me

)
Catch pan below of it's a bigger job.
The lab squirt bottle is really effective, and you can reach inaccessible places with a focused stream, without using much.
It washes the grunge downward and away, so you often don't need to touch it.
This is usually a very low mess, neat sort of cleanup.
A few square inch piece of low cost microfiber towel, pre-wetted with a squirt of mineral spirits will soak up an amazing amount of grunge before it's too nasty. I cut the towels into 6 pieces for grease/oil clean up. 2 or 3 of those pieces usually finishes up nice and clean, vs sheet after sheet of paper towel or blue Scott towels.
If it's just general engine bay cleanup, dust & mild road grime, then I use
Super Clean (something like
Simple Green), large soft brush, and hose down.
Odorless Mineral Spirits & fire danger - I tried my best to light up a puddle of this stuff. Couldn't do it.
Doesn't seem to cause any harm on the skin with short exposure, except maybe drying the skin.
However, if you wear rubber gloves, and gets under/inside while you continue to work for a while - uncomfortable prickly feeling that lasts a couple of hours after cleanup. I really try to avoid that now.
I have never seen Odorless Mineral Spirits damage any material.
I HAVE seen a water based cleaner permanently etch all the aluminum surfaces of a Subaru engine - never to look new again. Don't remember what kind it was.
If it's really nasty big job, say - the whole undercarriage, and I have access to one, then pressure washer is great.
But - nasty stuff gets all over the driveway, which then needs another cleanup of it's own.