I heard this sound sometimes when logging on servers in older versions, but I just had it happen on singleplayer. Basically, I'm not sure how to reproduce this, but very rarely when opening a world, you'll hear the 'minecart.base' sound. I'm really not sure how to reproduce it, but I put a video below of the issue in action. Maybe try to maybe make a world then duplicate it, then play on the copy, then delete the copy and redo. That's what I did and eventually it happened.
I'm not too sure but I think this glitch is much, much more common in the latest snapshot than in the past. In the past I believe I heard it probably only twice or three times, and I just had this glitch happen almost twice in a row.
I was lucky enough to get this rare glitch on video, here it is: https://youtu.be/qchGJDQlNys
(Please mind the long duration of the loading screen and the window being recorded on the wrong side. The recording software I use is laggy and buggy)
How to reproduce
Have subtitles enabled
Create world with seed -6781790025949582501
Observe the issue right after entering the world
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Not sure, but possibly related to MC-35714 ?

Is this still a concern in the latest Minecraft version 13w48b? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.
This ticket is incomplete without the requested information, no response has been received within a reasonable time and we are assuming the issue has been resolved. If you are still experiencing this issue, we can reopen it at your request.

Please reopen. I have this issue sometimes and it is very stange.
Please also update affected versions. Thank you.

Still an issue in 14w11b, can a mod please update the affected versions? Thank you.

Please reopen and add 14w11b as affected. Thank you. In the other report it is a Video https://bugs.mojang.com/browse/MC-38589

Reopened.

Maybe related to MC-36502


Entering a minecart still briefly plays the riding sound in 1.8-pre3.

Unable to reproduce so far in 1.8.

Then let's call this fixed for now.
About a day ago I had this bug happen again (MC-89729), so I say confirmed for 15w40b, please reopen.
Reopened and given the ticket to Spake Miner.
I'm still trying to reproduce it on 15w41b. It might take a bit of time, but if anyone can also try to get it and maybe confirm it, I'll be thankful.
Confirmed for 15w46a

Confirmed for 1.9.
Confirmed for 1.9.2
Confirmed for 1.10.2

Interestingly, in 16w32b, I cannot hear this. However, if I enable subtitles, then I see 'minecart rolls' in the bottom right. So the game is definitely trying to play it. Here's a video: https://www.youtube.com/watch?v=y1qa8DESjgM&feature=youtu.be

Confirmed for 1.11.
Confirmed for 1.11.2.
However, I wasn't aware this was a bug. I just assumed that whenever this happened, there was an abandoned mineshaft generating (which contain minecarts). However, I have noticed that I can't ever actually hear it, but just the subtitle is shown.

@unknown, 1.11.2 was already marked as affected
Affects 17w14a.
Addition to description:
How to reproduce
Have subtitles enabled
Create world with seed -6781790025949582501
Observe the issue right after entering the world
This still happens to me as of 2017.11.13 (playing on 1.12.1 with mods). Saw a comment somewhere mentioning this sound is caused by mineshafts in the spawn chunks.
I encountered this annoying issue when joining my single player world on Minecraft 1.12.2 and decided to do some debugging to try and figure out why this is happening, I've found two issue which might be causing this issue:
MovingSoundMinecart is played every time an entity which is an instance of EntityMinecart is spawned (spawnEntity method in the WorldClient class).
MovingSoundMinecart is played by the SoundManager/SoundSystem before it's update method is called meaning the sound position (XYZ) is 0 / 0 / 0.
I encouter same sound when I load a world, a freind of mine too, it not happening every time when a open the world, usually when I enter in game then in the world.
Discovered while joining my server on Pre6
Still an issue in 1.13-pre8

If you're still encountering this issue I will need your save file, as I am unable to otherwise reproduce.
I have this still occurring in 18w31a. I generated a completely new world and noticed "Minecart rolls" appearing in the subtitles when I moved beyond spawn and new chunks were generating.
A zip file of my world save has been attached. (The world was re-created. I tested a few times beforehand to make sure the issue persisted on a new generation, and it does.)

--> MC-91163
I tested several times your "18w31a testing" with subtitles : on and there are no subtitles "Minecart rolls" or the sound of the minecart that is played when I load the world.

"Minecart rolls" subtitle confirmed in a new 1.13 world.
that happens to me too, it freaked me out at first
The "minecart rolls" subtitle doesn't seem to occur on world load (at least not consistently.) If you travel outward to where chunks start generating, it appears.
The culprit is most likely mineshafts generating. While walking around the world in non generated chunks with subtitles on you can see "Minecart rolls". This makes it easy to tell if you found a mineshaft simply by running around and reading your subtitles.
Confirmed for 1.14, and it plays when standing still at my base.

Confirmed for 19w40a

Still occurs in 1.20.2. Possible reopen?
Maybe it's best to create a new report, seeing as it's been closed as fixed for several years already. It may be a reoccurance, rather than a "never was fixed".