What is the score that most classes use for class skills?
The initiative roll is based on Dexterity, and its modifiers. It is rolled before every combat by each participant. You will thus (in all probability) make more Dexterity checks by a raw count (due to how often all players roll for initiative) than any other single ability check. The bigger the party, the larger this single input goes toward skewing the results toward dexterity. (And your opponents also roll for initiative based on dexterity).
Using the notional 4 person party and the DMG recommended 6-8 encounters per adventure day, you have from 24-32 Dexterity checks alone before any other check/roll is made, plus a minimum of 6-8 checks by the opposing forces. If any of your party, or your whole party, is trying to be quiet as you approach an area, combat or not, you have a default addition of Dexterity(Stealth) checks added to that.
Experience: in five different campaigns, three of which are still ongoing, the dexterity(initiative) check is the single most common roll at all of our tables. (With Wisdom(perception), Dexterity(stealth), and Strength(athletics) following in that order ... that last one may be skewed a bit due to two games having had martial characters using the shield master bonus action shove a great deal.
At the beginning of every combat, you roll initiative by making a Dexterity check. Initiative determines the order of creatures’ turns in combat, as described in chapter 9.
(Basic Rules, page 64).
For other checks, it is going to depend a lot on how the campaign is run, and what the party's make up is, to see which checks the DM calls for the most. You can expect to see a lot of Perception, Stealth, and Investigation checks in any game when exploring, determining surprise and "do they hear/see us?" and finding stuff. Again, the density of a given check will be modified by what your party's make up is.
From page 61 of the Basic Rules, we find:
Acrobatics, Sleight of Hand, Stealth
Arcana, History, Investigation, Nature, Religion
Animal Handling, Insight, Medicine, Perception, Survival
Deception, Intimidation, Performance, Persuasion
Some examples of how this varies: if you run a party full of Rogues, you may find that sleight-of-hand and stealth checks (Dexterity)are very common. If you have a campaign focusing on social interactions, the Persuasion, Intimidation, and Deception rolls (Charisma) may be the most common.
We are in a campaign that has seen a lot of wilderness adventuring: Wisdom(Survival) checks are very common top avoid getting lost and for foraging (food), while as Wisdom(Perception) checks for noticing the beasts/monsters before they notice us. For this campaign, you could argue that Wisdom ability checks are the most common.
About "class skills"
Some classes get proficiency in some skills, but the way that skill / ability checks work in D&D 5e, anyone can make a check for any skill (there is an edge case for lock picking, noted here. You need thieves tools to pick a lock, but you don't have to be proficient with them). My 8 Charisma Barbarian can make an Charisma(Persuasion) check, though his odds may be lower than the Bard's. My Fighter in plate mail can make a Dexterity(Stealth) check, although his chances for success may be less than the Monk's.
Each ability covers a broad range of capabilities, including skills that a character or a monster can be proficient in. A skill represents a specific aspect of an ability score, and an individual’s proficiency in a skill demonstrates a focus on that aspect. (A character’s starting skill proficiencies are determined at character creation, and a monster’s skill proficiencies appear in the monster’s stat block.)
Backgrounds also offer skill proficiency
Skill proficiency is not solely determined by class; a PC's Background also has an impact on what skill proficiency they have.
For example, my Ranger has proficiency with thieve's tools, which he got from his Background of Criminal/Spy.
My warlock in my nephew's campaign has Performance as a skill proficiency, thanks to taking the Entertainer background.
That background offers proficiency in both Acrobatics and Performance, along with Tool Proficiencies in the Disguise kit and one type of musical instrument. My Warlock's cover story, when not in a dungeon or in the wilderness, was as a traveling minstrel / singer / troubador.