Some elements are so distinctively suggestive of independent armorial displays that there are rules that limit their use as part of a larger design.

Inescutcheons, cantons, pennons, and sails should not look like they are displaying secondary arms that the submittor has no right to display.
— [Artair MacArtair of Orkney, Return, May 1983 LoAR]

As noted above, this protection applies to single escutcheons, cantons, banners, and sails, but the rules are slightly different for each of these shapes, as I will catalog further below.

Basic Principles

The central idea of each of these rules is to avoid registering armory which suggests that the bearer is entitled to honors they have not earned.

SENA A.6.C. prohibits the use of a canton or a single escutcheon in any armory if it is charged or divided into multiple tinctures (although such items may be added later as an augmentation). This covers both fielded and fieldless armory, although the nature of cantons limits them to fielded armory; in a fieldless design, surely a canton would be merely a delf. This rule only affects cantons and escutcheons, not lozenges, roundels, etc.

SENA A.3.A.3. says that any augmentation which appears to be a display of independent armory must be evaluated as if the augmentation itself was an independent submission, including checking for conflict, etc. Although cantons and single escutcheons are used as examples in the text of the rule, the scope of the rule is not limited to them, and seems to also apply to lozenges, roundels, and any other charge which can be used as an independent form of armorial display.

In addition, the College requires that submissions which contain a flag or sail must have that individual charge checked for conflicts. This rule is not codified in SENA, but has been established by precedent, following a series of at-times contradictory rulings which seem to have established the conventions outlined below.

Rules By Shapes

Escutcheons

Escutcheons are protected as forms of display under several different principles:

• Escutcheons are the most recognizable form of armorial display, so any escutcheon appearing within a piece of armory yields an impression of “armory within armory.”
• In British heraldry, a man whose wife had inherited arms could place her family’s arms on an escutcheon in the center of his own armorial display, called an escutcheon of pretense. This display indicated a claim to being the head of that family, but only on a conditional basis; for example, were they to divorce, the wife’s family’s arms would leave with her.
• Some period monarchs displayed their personal arms on an escutcheon in the center of the arms of the territories which they ruled.
• Augmentations of honor granted by period monarchs to armigers sometimes took the form of escutcheons.

Therefore, the following rules apply to escutcheons used as armorial charges:

• Single escutcheons may only be included in devices and badges if they are uncharged and of a single tincture; multiple escutcheons do not have this limitation.
• Charged or divided escutcheons may be added as an augmentation, after conflict checking.
• All escutcheons must be checked for conflicts as independent armory. [Update, December 2020:] This probably only applies to single cantons, as noted below.

In period and modern heraldry, an individual may assert a claim to land or property by placing the armory associated with that property on an escutcheon in the middle of their existing armory. An augmentation of honor… occasionally… takes the form of a charged escutcheon. Therefore… a single escutcheon may be used in an armorial submission only if it is uncharged and of a single tincture. Multiple escutcheons do not have to follow this limitation.
— [SENA A.6.C.]

Due to conflict-checking rules, single plain ermine or vert escutcheons are not registrable.

This badge [Or, … a bear … sustaining an escutcheon … vert] is returned for conflict with the flag of Libya, Vert.
— [Barony of Aquaterra, Return, Dec 2019 LoAR]

[Update, December 2020:] There was a recent discussion as to whether a design with multiple escutcheons would require them to be conflict-checked in this way. The consensus was that answer was “no,” due to the explicit carve-out for multiple escutcheons in A.6.C.  This impression is supported by the registration without comment in June 2016 of a device that included “on a chief Or three escutcheons vert.”

Canton

Cantons are protected as forms of display under this principle:

• Augmentations of honor granted by period monarchs to armigers often took the form of escutcheons.

Therefore, the following rules apply to escutcheons used as armorial charges:

• Cantons may only be included in devices and badges if they are uncharged and of a single tincture.
• Charged or divided cantons may be added as an augmentation, after conflict checking.

An augmentation of honor often takes the form of a charged canton… Therefore, … a canton … may be used in an armorial submission only if it is uncharged and of a single tincture.
— [SENA A.6.C.]

I couldn’t find mentions of ermine cantons in any  LoAR, and the only submission with an uncharged vert canton was accepted in 1997, so I am not sure of whether future submissions of devices or fielded badges with uncharged plain cantons would have those cantons conflict-checked as independent armorial displays.

Banners and Flags

Banners and flags are protected as forms of display under this principle:

• Banners and flags are a common form of armorial display in period, and are the standard form of modern vexillological display, so a banner or flag that looks like armory appearing within a piece of armory yields an impression of “armory within armory.”

Therefore, the following rules apply to banners and flags used as armorial charges:

• All banners and flags must be checked for conflicts as independent armory.

… charged banners are checked for conflict against already registered armory…
— [Colin Tyndall de ffrayser, Return, May 1999 LoAR]

Due to conflict-checking rules, plain ermine or vert banners or flags are not registrable.

This device is returned for presumption through display of the flag of Libya, Vert. In past precedents, Libya’s iconic single-tinctured flag has been given less protection than the other single-tinctured design we protect, the arms of Brittany, Ermine. Libya’s protections are dismissed when they appear as part of a potential quartering, where Brittany is protected. … However, banners and pennons are still considered means of heraldic display. And given that the banners hanging above the sails are rectangular and wider than they are tall, they appear to be modernly shaped flags. In other words, the galleon is literally flying the flag of Libya. If any protection for this design is to be afforded at all, it must surely be in the form in which it was actually used, as a modern national flag.
— [Muirghen MacQuharrie, Return, Jul 2018 LoAR]

Additionally, flags and banners should not be held by animate charges as a fieldless badge in order to avoid the impression of registering a supporter.

… the badge [A wolf passant argent, collared and sustaining a flagstaff sable flying a banner of Gules, three trilliums argent barbed and seeded vert] has the appearance of being a supporter. The College of Arms neither protects nor regulates the use of crests or supporters, and therefore will not register any submission that appears to be one. … Supporters aren’t defined by posture, but by function. If a figure is holding up a display of armory then that figure is a supporter. …
— [Kingdom of Ealdormere, Return, July 2005 LoAR]

Sails

Sails are protected as forms of display under this principle:

• Sails in period were often painted with the arms of their owner or a badge indicating the nation which commanded them, and a sail that looks like armory appearing within a piece of armory yields an impression of “armory within armory.”

Therefore, the following rules apply to sails used as armorial charges:

• All sails must be checked for conflicts as independent armory, unless they are uncharged and of a single tincture.

… a charged sail would not appear to be an inescutcheon of pretense. However, this does not negate the research in the previous precedents (and supported by the College of Arms when they commented on this submission) which showed that charged sails appear to be independent displays of armory. … The most obvious analogous case is that of a flag or banner used as an armorial element. … Therefore, it seems appropriate to rule that a charged sail must be checked for conflict against already registered armory.
— [Eiríkr Mjǫksiglandi Sigurðarson, Returned, Jun 2003] 

Sails which are uncharged and of a single tincture do not need to be conflict-checked:

We have long held that a sail is not a display of armory if it is uncharged and of a single tincture. This [Argent, a sail vert] then is not presumptuous of the flag of Libya, Vert.
— [Morgan Grey Beard, Accepted, Jan 2014 LoAR]

If the sail is attached to a ship, and the entire ship-and-sail charge is counterchanged over a divided field, this might not create an impression of a divided sail which needed to be conflict-checked as such:

… the precedents and rules are in place to guard against the appearance of an independent display of armory. In this case, as the entire drakkar is also counterchanged in the same manner as the sail, this greatly reduces the appearance of the sail as a form of armorial display. A sail is not considered a form of armorial display if it is uncharged and of a single tincture, or uncharged and of a multiple tincture shared by the rest of the ship.
— [Evan Hardrada, Accepted, Dec 2012 LoAR] [OSCAR]

Any Form of Display Used as an Augmentation

Any augmentation which consists of a charge which can appear to be an armorial display must be conflict-checked as if it was independent armory.

An augmentation that appears to be a display of independent armory, such as a charged canton or a single charged escutcheon, must also be evaluated as if the augmentation itself were a submission of independent armory for purposes of style, conflict, offense, and presumption.
— [SENA A.3.A.3.]

All Other Charges

No other charges are subject to these rules.

It’s worth calling out chiefs in particular, in part because the use of chiefs of allegiance is so common in period.

A chief is not generally considered an independent display of armory, and so it needn’t be considered as such.
— [Duncan Bruce of Logan, Accepted, Mar 2012 LoAR]