There is a more recent version of the Play Store available for download here: Play Store v5.9 Begins Preparation For Android 6.0, Adds Support For Fingerprint Readers, An Uninstall Manager, And More [APK Teardown + Download]

And a few earlier ones:

Google is in the process of rolling out a Play Store update v3.7.15 which follows 3.7.11 and 3.7.13 from early July. The latest version doesn't bring any new features as far as I can tell, so we can only assume it contains under-the-hood improvements and/or bug fixes.

One such bug fix resolves the problem with Sprint devices that were previously getting stuck downloading applications on 4G and displayed a "waiting for network" message.

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If you haven't gotten the 3.7.15 Play Store update automatically yet, you can find a number of download mirrors below. Just push the apk to your device and install it the same way you would sideload any other app. The file is signed by Google and was verified as legitimate (in fact, I pulled it off my phone a few minutes ago).

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Download

The APK is signed by Google and upgrades your existing Play Store. The cryptographic signature guarantees that the file is safe to install and was not tampered with in any way. Rather than wait for Google to push this download to your devices, which can take weeks, download and install it just like any other APK and enjoy the new Play Store experience.

Thanks, @_ERib!