There are four types of movement speeds outside of the normal movement speed that everyone has (even if it's specified as 0 ft. for some creatures), and those are as follows: burrow, climb, fly and swim (MM, p. 8).
Here are some examples of how a PC can get these speeds. These lists are not exhaustive, they're just everything I can remember off the top of my head. Also note that these page numbers are for the feature that grants the speed, not the overall race/subclass.
It's possible for a PC to have a climb speed; they can be a Tabaxi (via Cat's Claws; VGtM, p. 115) for example (I was going to include the Thief Rogue feature Second-Story Work, but that doesn't actually give you a climb speed, it just says that climbing costs no extra movement).
It's possible for a PC to have a fly speed; they can cast fly (PHB, p. 243), be an Aarakocra (EEPC, p. 5) or Winged Tiefling (SCAG, p. 118), or gain wings as a Draconic sorcerer (via Dragon Wings; PHB, p. 103) or Divine Soul sorcerer (via Otherworldly Wings; XGtE, p 50).
It's possible for a PC to have a swim speed; by being a Water Genasi (EEPC, p. 10), Lizardfolk (VGtM, p. 113), Triton (VGtM p. 118), Sea Elf (MToF, p. 62) or by getting the "Gift of the Depths" Eldritch Invocation (XGtE, p. 57).
But is it possible for a PC to have a burrow speed, either temporarily via a spell (such as how it's possible for a creature to gain a fly speed via the spell fly) or permanently, either via a racial feature or something gained later on (feats, class features, magic items, etc)?
I know of no RAW or even UA races that have a natural burrow speed, nor do I know of any spells such as fly that grant a burrow speed. But there might be something I'm missing.
So, to restate my question: Is there a way for a PC to have a burrow speed?
Some things that don't count: anything homebrew (UA or other playtest material is fine, though), or Wild Shaping, Polymorphing or otherwise becoming a different creature that has an innate burrow speed (I want the PC to still be the same creature, even if the burrow speed itself is temporary).
The context for this question, by the way, is that I plan on giving a burrow speed to a homebrew playable race and wanted to know if there's a precedent for it already so I can gauge how powerful it would be. However, I still want the focus of this question to be on existing methods by which a PC can gain a burrow speed.
Best Answer
Try claws of the umber hulk.
Appearing in Princes of the Apocalypse, p. 222, claws of the umber hulk are a rare magic item granting (among other things) "a burrowing speed of 20 feet."
To the extent it is helpful for gauging "how powerful" they are, note that the claws come with some significant pros and cons. On one hand, they give the wearer a melee attack that deals 1d8 slashing damage. On the other hand, they require attunement, and while wearing them "you can’t manipulate objects or cast spells with somatic components." That is a nontrivial downside for many PCs, and for some -- like spellcasters -- a probable deal-breaker.