Up until about 1800, women wore garments that were known as (a pair of) stays or (a pair of) bodies. The term "corset" only came in later. Today the terms are often used as synonyms, but I (and most re-enactors) use the terms "stays" and "corset" to differentiate between the pre-1800 and the post-1800 shape. The following is true for both stays and corsets.
For any woman who wants to wear historical costumes, stays are a necessity to achieve the proper body shape - the best of costumes wouldn't look authentic without. The question often arises whether that isn't awfully unhealthy.
For those who want a quick'n'dirty answer: No, it isn't. If you want to know why, read on.
Are stays uncomfortable? Yes and no. Yes, because they are more restricting than modern clothing. A corset forces you into an upright posture, which may be painful for our modern slouch-backs at first, but only because we're not used to it. You soon get used to it. It's difficult to bend down, but then it's unladylike to do so anyway. Picking something up from the ground is what gentlemen are for, and if none is available, bend at the knee as grandma was taught to do.
So a corset is indeed uncomfortable compared to the freedom of movement that modern clothing offers, but then you can't waer historical clothing and expect it to be comfy. Either you have the cake or you eat it.
As for unhealthy: Of course there were always ladies who overdid the waist cult. There were also parents who laced their daughters (and sometimes their sons) from an early age on. It was commonly believed that the as yet soft skeleton needed support in order to keep the body upright, so babies were often wrapped like mummies - sometimes so tightly that they died (Junker/Stille, 1991). Apparently, the connection between "soft skeleton" and "easily distorted" was not made. Or maybe they dodn't mind.
At a later stage, children of both sexes were made to wear cord-stiffened stays to achieve an upright posture, and to lay the basis for a narrow waist for the girls. Boys grew out of the stays at the age of 5-9, but girls "grew up" to get their first whalebone-stiffened set at the age of 10-14. It was a rite of passage, its only halfway appropriate comparison today being the first menses. At that age, the bones still were relatively pliable, so a certain degree of moulding, conforming to the current fashion, was to be expected.
By the way, all of this is only true for the upper classes. Lacing took time, and even back then, time was money. Many from the working classes couldn't afford stays as such, nor could they afford to bodily restrict themselves so much that they couldn't work properly.
Let's go back to the question of why the admittedly bad consequences of lacing don't have to concern us nowadays. It's really quite simple: We had the privlilege of growing up without lacing so that our skeletons could take on the shape they were meant to. An adult skeleton is quite literally hard as bone, i.e. it can't as easily be deformed as a child's. Like a tree, it's flexible while it's young and keeps the shape it's bent into (ask any bonsai grower), but from a certain age on, you can't permanently deform a branch without breaking it. I should stress at this point that, despite what you may have heard, it's impossible to break a rib just by lacing tightly, unless you use heavy machinery. The worst thing that can happen to an adult woman in real life is sore spots due to rubbing.
Mind you, all the above is only true if you lace up only a few times per year. If you wore a corset every day, the muscles that keep you upright would atrophy like an arm in a plaster cast.
Pre-19th century stays are characterised by straight lines, i.e. they're funnel-shaped. Tight-lacing a skinny person in stays would have to compress the ribs, which tends to hurt, more so when the stays are removed than while wearing them. Nevertheless, the ribcage doesn't suffer any permanent damage. If the stays hurt, try lacing less tightly. You many have to employ two small half-moon-shaped pillows stuffed under the breasts to prevent them from dropping down into the loosened stays. Roughly the same goes for 19th century corsets.
As for children and teen-agers, corded stays were originally used. They should not be a problem if worn only a few days per year, especially if they're laced just as tightly as necessary to prevent them from chafing. If the child doesn't complain, it's OK. Just make sure that the kid understands that it's all right to complain if it hurts - some children think that they have to cope so's to please the parents. A teen-ager can wear lightly boned stays or corsets, and from the age of about 16-17, a proper corset should be OK - again, only if worn occasionally. I think that the appropriate kind of stays, worn on up to 10 days per year, should not cause more damage than the varying fashions of laying a baby either always on the back or always on the belly.