I come from playing Pathfinder, which has a laundry list of mundane arrow varieties you can stick in your quiver. Is there an equivalent list of arrows in 5E? If so, where can I find it?
There are 2 types of "special," non-magical arrows in the core rules: silver arrows and adamantine arrows. These arrows overcome the damage resistance and/or damage immunity of the target. (Arrows aren't special here, the above is true for daggers, etc.)
There's no list, exactly, but here is where you'd find them:
Silvered weapons are described p. 148 of the Players Handbook. Silvered weapons are expensive but generally available.
Adamantine isn't mentioned in the Players Handbook, and there's no "set price" for them. Although Adamantine Armor is a magic item, the (DM Guide, p 150) the Monster Manual mentions "nonmagical weapons that aren't adamantine" (MM, p. 140, etc.) in several places. It's left to the DM as to how these might be acquired.
There are no other special arrows, like whistling arrows, etc., specified in the core rules. (But the rules don't exclude them either, so as ever, talk to your DM.)
Update: Xanathar’s Guide to Everything, published since this answer was edited, lists a price for adamantine weapons as 500GP + the value of the normal item.
The adamantine version of a melee weapon, or of ten pieces of ammunition costs, 500 gp more than the normal version, whether the weapon or ammunition is made of the metal or coated with it.(page 75)
The simple answer is no. 5e ranged weapons use standardized ammunition varieties by default, meaning there is one "arrow" under the ammunition category, the damage and range of which is entirely dependent on the bow from which it is fired. Arrow variants designed for utility would need to be improvised.
If you use a poison vial, you can apply poison to your arrows.
If you fight something with paralysis or other types of stuff like that from naturally occurring fluids on the animal/monster you could augment some of your arrows if you are proficient with a poisoner's kit, though you'd have extract and store the liquid and make the vials in your downtime. Also if you fail to extract the poison safely you get poisoned.
This post has more on Poisoner's Kit: How does crafting poisons work?
There are none already in D&D 5e, so adding these kinds of arrows is at the DM's discretion. (This freedom to invent things is why I find RPGs so much fun.)
Talk to your DM and agree on something that works for your group. In our own game I argued that a high-end armory might sell a bodkin arrow (made to pierce plate armor). We agreed on 1 gold per arrow (vs. the 1 gold for 20 normal arrows listed in the PHB), and each arrow has +1 to hit against armored or naturally armored foes. Not completely overpowered, but still a useful, if potentially expensive, option.