I think a lot depends on age and if they are already sleep trained or not.
Mine now goes to bed with a music CD on (same CD every night for years...well we did change it once and that stuck for a few months before returning to the original) and 3 muslin squares in his bed. He sniffs the muslins and rubs them on his face and holds and rubs the wash label (silky ribbony bit) between his fingers. But he is 5yo. It was not the same when he was a baby.
I see on another of your posts that your LO is about 8 wks old.
I can't say for certain when I started sleep training. Either day 1 because I always had in mind that I would help him as much as needed but I really wanted to put him down (sounds like I didn't want to hold him, I did, but my character is such that I knew I would fail if I attempted to hold all the way through A, all the way through E and all the way through S, I would have crumbled), I started out with Harvey Karps 5S wind down routine as I had not heard of BW until DS was 4 weeks old. Then I used a mix of Karps method with Tracy's BW method. So, maybe I was aiming for sleep training and self soothing from day 1 or maybe it was from wk 4 or maybe it was whilst I was still pregnant as I used to sing one song repeatedly throughout my pregnancy and DS recognised it and calmed instantly on day 1 when I sang it to him.
...somewhere between say 6 - 8wks he was self settling. A strong wind down routine with cues for sleep and if he needed additional help for some sleeps he got it but on the whole I'd say he was independently sleeping - he refused to sleep in arms.
In those early weeks he did most of his falling to sleep with his eyes open, I stuck strong black and white patterns to the side of this cot to stare at (just simple computer printed wide lines), I could see him do his "7 mile stare" which Tracy describes in her books. then he'd nod and fall to sleep. His wind down included swaddling, the key song whilst being held with a muslin square on my shoulder so it was by his face, shushing and slow dancing (or rocking in arms and back rubbing rather than patting) then a key phrase and into his bed. He would stare, then nod, then sleep.
I stopped swaddling at 10 wks just swaddled for the wind down if he needed it then unswaddle to go into bed, WD always had the muslin square (so did E time and A time it was a constant). He was too young to leave a muslin square with him it would not have been safe and he would not have had control of it. If he did need more help I put a firm hand on him and rocked a little whilst he was in bed. This was also the time he stopped having the black and white images to stare at, he stared at his cot side instead.
I am not totally sure when he began to mantra-cry. Possibly not in the very early days I don't remember. I do remember later hearing it. so he would kind of hum repeatedly. When very tired it sounded sad but there were no tears, it was not a real cry and he did not want me with him.
When a bit older I let him fall to sleep with the muslin square near his head then removed after he fell asleep.
Older still and seeing how he could control the muslin I let him keep it. By then (6 months) he could hold it and fling it around, wave it about, rub his face with it, look like he was frantically banging it around but this was part of his self soothing. By about this age I knew if his mantra went on for more than say 5 mins he needed extra help, I'd pop back in to him put a hand on for a few mins and this helped him relax. But it is normal for a mantra to go on for the full 20 mins of falling to sleep, just that for mine I knew he wasn't going to manage alone.
Self settling is quite individual I think but can often not look or sound "relaxing" in the adult sense. The mantra and all the lovey flinging can look totally un-relaxing but many of them do this type of thing and then nod off so it must be relaxing for a baby
