While this question focuses on character abilities, I will expand my answer to abilities outside of what a character has. Particularly abilities, or effects from items.
The sage advice compendium is a compendium of official rulings. It provides a test for determining whether a game feature is magical
Is it a magic item?
Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description?
Is it a spell attack?
Is it fueled by the use of spell slots?
Does its description say it’s magical?
According to the sage advice compendium, if a game feature meets any of these points, it is magical. The compendium then explains a situation where none of these points are met, and concludes that the feature is not magical; so, even if it isn't explicitly stated, it is implied that if these points aren't met, then the feature should be considered not magical.
With that said, lets add a list of abilities that come from items which are not considered magical:
- The ring of shooting stars, is a magic item, it is considered magical, it would thus be rendered mundane in an anti-magic field, however it has the following ability:
- "You can expend 1 to 3 Charges as an action. For every charge you
expend, you launch a glowing mote of light from the ring at a point
you can see within 60 feet of you. Each creature within a 15-foot
cube originating from that point is showered in sparks and must make
a DC 15 Dexterity saving throw. taking 5d4 fire damage on a failed
save, or half as much damage on a successful one.
This ability is not magical, the ability is not a magic item, it is not a spell, it doesn't make a spell attack, it isn't fueled by the use of spell slots, its description doesn't say it is magical. This ability could extend into an anti-magic field while the item is outside.
Chime of opening, I'll spare the text of what it does, it is a magical item itself, its ability is not a magic item, nor does it meet any of the other points, so it is not magical.
Horn of silent alarm, another magical item with a non-magical effect.
Devastation orb, magic item, but 3 of the 4 elemental orbs' effects don't check any of the boxes, and thus using the SA compendium, are not magical effects, and would apparently reach into an anti-magic zone.
every item ability that doesn't specify it allows you to cast a spell, create the effect of a spell, or make a magic attack. There isn't a single magic item I could find that has an ability that is fueled by the use of spell slots, or says it is magical in its description, and the magic items' effects are not themselves magic items. Thus every ability from a magical item is not a magical ability unless the ability is to cast a spell. Basically it is rare for a magical item to have a magical ability
Disclaimer: As I have pointed out, following the SA compendium, rarely do magic items have magical abilities or effects. Every time I have pointed this out, however, I have gotten a lot of flak from the community I have pointed it out to. I myself have the opinion that what I have pointed out is indeed not RAI, even though it is the official ruling.
There is strong evidence that Jeremy Crawford doesn't rule this way (even though he technically wrote the sage advice compendium). For example, here is a tweet which say the following:
Light from any magical source can illuminate the area of a darkness spell, but the darkness spell can dispel light created by a spell of 2nd level or lower, not light created by a non-spell.
In this tweet JC's wording would indicate that light from a magical source would illuminate the darkness spell, this would require that the light is magical. A magic item giving off light would make it a source of the light, it is a magic item so it passes the compendium test for being magical, so it is a magical source. Thus using JC's wording from this tweet, any light from any magical item is magical light, even though almost all light from magic items would fail the Sage Advice Compendium Test.
Using this tweet as the basis, if any light from any magical source is considered magical I think we can add a new qualifier for being magical to the sage advice compendium. If an effect comes from a magical source, that is a magical effect.