While any class can take any background and there are obviously options - there are no options that really boost Martial characters the way that the ostensibly Martial backgrounds boost non-martial characters.
Rogue's are technically Martial classes, but this post is focused on Fighters, Paladins, Barbarians, Rangers & their Subclasses
SELLSWORD: Would appear to be a background that could "fit" someone's concept of a Martial character.
However none of the benefits truly add value to a Martial character such as a Fighter or Paladin.
SOLDIERING: Athletics and Intimidation are the granted proficiencies. The problem is that Martial classes often have very narrow lists of Proficiences, and both of these are already on it (except for Rangers who do not have Intimidation as one of their options). So these two "bonus" options do not expand beyond what most of the Martial Characters can already get.
ARMORED: Proficient with Medium Armor. This adds no value to any of the Martial classes (Fighters, Paladins, Barbarians, Rangers) who all already have at least Light & Medium Armor Proficiency.
LAWKEEPER: Another Background that would seem very suitable to a Martial Character concept
DETECTIVE: Perception & Intimidation. Slightly better than Sellsword here, as Perception is not on the Paladin list and Intimidation is not on the Ranger list.
ENFORCER: Proficient with Martial Weapons. Again, as with "Armored" above this is basically useless for Martial Characters.
Being able to add some of these options to Non-martial characters are great - especially Armored & Enforcer. Yet, for Martial Characters these backgrounds add very little if anything.
On the whole, I think backgrounds need a look over to make sure they all add similar value and that there are options for all classes that can expand on what they get in their base class.