Simply inserting a genuine flashdrive (as opposed to an attack device that mimics the appearance of a flashdrive) shouldn't ever be a risk. Similarly, browsing the files on a flashdrive, copying them to your local machine (as long as you don't put them somewhere where other programs will expect to find a trusted file), or writing to the flashdrive should always be safe. However, files on the flashdrive should be considered untrusted if somebody else has accessed the device; for example, if there was an executable on the drive its contents could have been replaced with a malicious file but the name, size, timestamp, and other metadata have remained the same. If that happened, running that executable would of course compromise your system. Opening non-executable files from an untrusted flashdrive could also be a risk, depending on how secure the program that opens them is and how much it trusts the files.
Now, with that said, there has been malware that spread by flashdrive before, but only on old or unpatched operating systems. Earlier versions of Windows could be configured to automatically run a program on a flashdrive or CD-ROM, but that no longer happens by default. Security vulnerabilities in how Windows Explorer generated icons when browsing a directory were used to infect computers from flashdrives in at least one notable case (Stuxnet), but that bug was patched long ago. It is not impossible that some other vulnerability may have been discovered, but so long as your machine is running an up-to-date OS and file browser, and you don't open any untrusted files, you should be fine.