How commonly were shields carried by knights and men-at-arms in the 14th century?

You'll have to narrow that down quite a bit. :) That century saw a great deal of change and what may have been popular in one quarter of the century (perhaps even in one decade) may have been passe the next.

Well by the turn of the century, 1400 AD, were knights still commonly using shields in battle? I know by the end of the 15th shields had kinda gone out of mounted combat.

They could have been, although shields stop showing up in (English) effigies shortly after 1350.
They were certianly still used for tournaments, though.
The Black Prince shield dates from probably 1370-1375.

as plate armour comes in, shields go out, but that's a vast generalization. I did a paper in University (25 years ago) on shield use in the 14th c., so I'll synopsize.

1) While the idea seems to be that plate armour made shields unnecessary, what you see is that lesser men at arms and even Irish kerns and other less armoured men also seem to use the shield less, at least in contemproary illustrations.

2) While the men still carrying shields seem mostly to be mounted, the assertion that the shield was for resisting the lance doesn't stand up to, for instance, Fiore's mounted combat plays, http://www.hoplologia.org/fiori/41recto.html wherein men engaged in lance vs lance have shields, but men engaged in swordplay even vs a lance do not. It would seem that men discarded shields when their lances broke--or even didn't carry them.

3) As knights increasingly used two handed weapons on foot, and fought on foot (polearms, pole hammers, and long swords) they didn't use the shield. This may be part of a vicious technological circle that includes the armour--they needed heavier weapons to beat down opposing plate armour, so they didn't bother with a shield to give two hands for bashing. ???.

4)But... I was able to find fifty or more references to English men at arms with shields. As one example, when a squire--can't remember who--is knighted at Poitiers, his shield was immediately marked with his arms--that's in Vie de Price Noir, I believe. So he obviously had a shield. BUT did he carry it in battle? Did the better quality men at arms have a "tool kit" of weapons and armour suitable for different combat conditions? (I'm mounted on a sunny day vs archery--give me my heavy helm and that nice big shield, please, squire?)

5) Finally, against a man trained the way Fiore suggests knights were trained, a man with a shield would really be at a disadvantage--hate to face a trained wrestler when I'm trapped behind a heater. As we've learned in my combat group doing Ancient Greek stuff, a clever opponent can break your shield arm with your shield. So the decline in shield use may be due to a new training system--not that there's any evidence that Fiore's system was Europe wide.

That was a long winded way of saying--I don't think anyone knows. By 1450, virtually no one outside a tournament has a shield. In 1350, I think most gens'darmes at least had one in the baggage.

Hope that helps.