While your note about chainmail under plate armour was informative, I can’t help but wincingly imagine that people sometimes got hit really hard in those flexible places and got rings of chainmail buried in their skin. Do we know much about what kind of injuries knights and similarly armoured fighters sustained in the course of battle?

Well, people did wear heavy quilted gambesons between the chainmail and their skin, but yes, having chainmail rings driven into someone’s flesh absolutely could and did happen. 

As to what kinds of injuries could happen, let me introduce you to a favorite of medieval surgical manuals, the Wound Man:

The Wound Man was supposed to teach surgeons and other medical professionals all the different ways that someone could be injured. To take a quick inventory by body part:

• Head: hit by a club, hit by a rock, stabbed by a poinard/misericorde, stabbed by a knife.
• Shoulders/arms: slashed by a saber, hit by a rock, shoulder broken by a hammer, elbow broken by a club, hand cut off by a cannonball, hand partially cut off.
• Torso: shot with an arrow, stabbed by a javelin, run through with a sword, stabbed by a poinard/misericorde.
• Legs: boils, shot with an arrow with the shaft intact, arrowhead left embedded after shaft breaks, leg pierced by a spear, foot stabbed with a spear, treading on a thorn, leg broken by a cannonball.