My experience over the years with Walschaerts, Stephenson, Baker, and Southern, along with what I have read have led me to the following conclusions. Not to be taken as Gospel.
Walschaerts has the advantage of being relatively light, has small bearings where tolerances can be kept pretty tight, not too many moving parts, and it's on the outside of the frame.
Stephenson's main advantage is that it has variable lead which gives slightly better performance through the speed range as the gear is hooked up. Because of its geometry the gear causes the lead to increase as the link block gets closer to center which usually occurs at higher speeds. More steam earlier gives better cushion. Less lead starting should mean less chance of getting stuck on center, though that has not been my personal experience. The disadvantage is that the parts are very large, even on small locomotives. In the USA, it was between the frames, and it had lots of places where things could come apart. The clearance necessary on the large diameter eccentrics invariably induces different timing events at different speeds because of the mass and speed of rotation.
Baker is weird. It gives uneven extreme valve travel, but I have not seen it be a detriment in our operation. The advantage is that there is not a sliding link block to wear. With needle bearings the gear could have virtually no slop. There is some leverage disadvantage that would lead to pretty high stresses.
Southern seems pretty cool. Only experienced it on EBT 17, and despite some other deficiencies in the locomotive, I thought the gear worked very well. It, like the Baker takes vertical motion and transforms it to horizontal motion. The advantage over Walschaerts is that the link block only slides when the quadrant is moved. With Walschaerts, the link block is always moving in the link except in the case where the hanger length and the rotation radius of the link are equal. This occurs in only one direction and one location.
