My knowledge of D&D is primarily 3.5. In that edition, there were many modifiers that affect AC, Natural Armor, Deflection Modifier, Dodge Bonus, Insight Bonus, Luck Bonus, and Sacred Bonus. I have not been able to find any of these rules in any core rulebook. My question is whether or not these are still official rules?
5e doesn't have modifier types
Though most modifiers in 3/3.5e were assigned a specific type and would not stack with other modifiers of the same type, 5e does not have this concept of modifier typing. Modifiers in 5e are effectively untyped, and the only rule about stacking is that bonuses provided by the same source do not stack, as explained by the DMG:
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. [...] Game features include spells, class features, feats, racial traits, monster abilities, and magic items.
So you can't double an AC bonus by casting the same spell on yourself twice, for instance, but if you are affected by two different spells that both give you an AC bonus, they stack together.
However, there is a caveat. Many effects in 5e confer improved AC by allowing the use of an alternative formula to calculate your base AC, rather than modifying your existing AC - most notably, this is now how wearing armour works. However, as described in the PHB and Basic Rules:
Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.
Since you must choose one formula to calculate your AC even if effects mean you have a few choices available to you, this is a way of preventing some stacking effects. Mostly it seems effects which would have provided Armour or Natural Armour bonuses in 3.5 now work in this way.
In general in 5e it's harder to get as many bonuses going at once to stack with, because most useful magic items require attunement; typing the bonuses and preventing them from stacking as well would be overkill. This is part of the general simplification of 5e; you might also note (if you haven't already) that 5e has no such thing as a touch or flat-footed AC, where the types of AC bonuses you had were very relevant.
Yes, there are plenty
You can stack many distinct sources of AC on top of one another. There are many, many modifiers of AC in 5e.
Base Armor Class: This is 10 + Dex modifier (unarmored), or a different formula if wearing armor or granted by a class feature. This is all you really need to know coming in new to 5e, but it doesn't stop here
Shield: +2 AC, can only benefit from one shield at a time
Magic armor: Can range from +1 (uncommon) to +4 (legendary) per item, so you can have a +4 shield and wear a +4 armor. Note that bounded accuracy in 5e means a +4 bonus from any item should be so legendary, it probably shouldn't exist
Magic trinkets: There are rings of protection for +1 AC, cloak of protection for +1 AC, or ioun stone, protection for +1 AC. You also have the animated shield for +2 AC and the Defender sword for between +1 to +3 AC
Spells and potions: Haste grants +2 AC for 1 minute, shield of faith grants +2 AC for 10 minutes, and the shield spell grants +5 AC for 1 round (6 seconds)
Class features and optional feats:
The Dual Wielder feat grants +1 AC when both your hands have weapons, but it means you can't use a shield
Defensive Duelist allows you to add your proficiency bonus to your AC against one attack
The Defense fighting style gives +1 AC when wearing armor
The Mariner fighting style, from the Unearthed Arcana, gives +1 AC when wearing light, medium, or no armor
The Battlemaster Fighter's Evasive Footwork allows you to add 1d8 to your AC while moving during your turn
A Bladesinger Wizard (Sword Coast Adventurer's Guide) can use bladesong to add their Intelligence modifier to their AC for 1 minute
A Wild Magic Sorcerer's surge result can give you a spectral shield that gives +2 AC
A Ranger's Multiattack Defense lets you add +4 AC against one creature's attacks against you, starting with their 2nd attack
Spellcaster magic items:
A Robe of the Archmagi sets your AC to 15 + Dex
A Staff of Power grants +2 AC
Blessings: The Blessing of Protection grants +1 AC
Situational Modifiers: You can add +5 to your AC when you have 3/4 cover, or +2 when you have 1/2 cover
Such categories are not defined in 5e and for most creatures and situations only the armor it wears and its Dexterity have an effect on AC.
Spells and abilites can still provide boons to AC, but the rules for these are detailed in the description of the particular effect. They are not grouped mechanically and thus there are no rules that would effect a given subset of them as such.