I understand that the family motto appears below the escutcheon on a coat of arms. While I've found a couple of (somewhat-different) escutcheons for my family -- Field -- I've not been able to find a family motto. Does anyone have any ideas of what it might be, or where I could find out, please?
There is no such thing as a family coat of arms (or whatever). Arms are granted to specific people and may then descend to eldest sons etc. Anyone else with that surname is not legally allowed to carry those arms. Hence your different escutcheons will have been granted to different people and have nothing to do with each other.
This refers only to the United Kingdom.
The one exception in the UK is the family crests in Scotland which are explicitly allowed to be used by all members of that clan. Even there, there are specific rules about which variation of the crest can be used by ordinary clan members.
Picking up on the Original Poster's description of the Field coat-of-arms that he's seen ("sable, a chevron engraved between three garbs argent"), this is the escutcheon used by "Field of Laceby and Ulceby" according to volume 55 of "Lincolnshire Pedigrees" - specifically "Sable, a chevron engrailed between three garbs argent". Note that the term is "engrailed", not "engraved" (personal note: no, I didn't know either). Engrailing is effectively a scalloped edge with the points of the chevron pointing out into the surrounding background.
Burke's Landed Gentry of Great Britain & Ireland (1886) (digital copy in my possession) has two Field families, viz: "Field of Laceby" and "Field of Ashurst Park". Unfortunately neither describe a coat of arms.
The Coat of Arms Database site (which, to give it credit, does explain "The term family crest is a misnomer. For the most part, arms belonged to individuals and not families or surnames") has 9 different coats of arms for the name Field. The first three appear to be variants on "Field of Laceby and Ulceby" referred to above, differing only in the crest (or not). The others are radically different. None show any trace of a motto.
Whether or not the College of Arms would permit both "sable, a chevron engraved between three garbs argent" and "sable, a chevron between three garbs argent" (i.e. a plain chevron) I have no idea, but it must be distinctly possible that this is an error that's crept in.
As for mottoes, the chapter on mottoes in "A Complete Guide to Heraldry" (of 1909) appears to suggest that in England & Wales (but not in Scotland), mottoes are just a personal add-on to the arms, are not hereditary, are not defined by the Letters Patent creating the arms and it's entirely up to the person with the arms what they use and how.
The official keeper of the history of coats of arms in Britain is the College of Arms.
They have an enquiry service and would be the ultimate reference. Anything beyond a simple enquiry is liable to attract a fee, so check with them before committing to anything.
Don't forget, not all arms have a motto. Although I believe the great majority do.