The Nexus 4 & 10 are NOT compatible with Mifare Classic tags because Mifare Classic tags do NOT adhere to the NFC Forum's Standards. They can not write to the tags nor can they read information that someone else has written to the tags. However, they can detect the UID Code (kinda like a UPC code) of a tag which is unique to every tag. So, if you use an app like NFC ReTag or NFC Task Launcher which has the ability to just detect a tag and read it's UID code, and then launch settings/profiles/etc then you can still use Mifare Classic tags with the Nexus 4 & 10. This is why some people keep saying they are compatible. Just remember, technically they are NOT compatible - the phone can just read the UID code off of the tag.
What kind of tags DO work with the Nexus 4 & 10? Any tags that do adhere to the NFC Forum's standards. The most popular of which is the NTAG203. Unlike the 1K Mifare Classic which has 700 bytes of usable memory and is fairly inexpensive, the NTAG203 tags have about 140 bytes of usable memory and are slightly more expensive in general. However, 140 bytes is plenty of memory for most settings/tasks launching NFC Apps. Many apps, such as NFC Smart Q, allow you to create tasks and see how much memory is needed even without having tags so you can do that before ordering tags to make sure it's enough. The only things that it might not be enough for are vCards or if for some reason you want to program a really long text string to a tag (like more than 130 characters); but since most of us want to use tags to automate things, 140 bytes is enough for that.
What problems might you encounter using Mifare Classic tags to trigger events using the tag's UID? If you only use one app that detects a "blank" tag (Mifare Classic's are seen as blank by the Nexus 4/10) and read's its UID code and triggers a set of rules/settings/tasks based on that then you shouldn't have any problems at all! However, if you have more than one app that detects blank tags, then anytime you tap a Mifare Classic tag, you'll get a pop-up box asking you which app you want to use to execute the action which defeats the purpose of automation.
What is the deal anyway? Why don't the Mifare Classic tags work with the Nexus 4/10 like they do with all other Android phones? Basically, NXP is one of the leading manufacturers of NFC products. They not only manufactured the NFC hardware built into pretty much every Android phone out there up to this point, but also manufactured most of the popular NFC Tags in use. At some point they designed the Mifare Classic NFC Chips and designed them specifically to be compatible with their NXP hardware, but did not design them according to the NFC Forum standards which meant they wouldn't necessarily be compatible with all NFC hardware by other companies. Since pretty much all Android phones used the NXP NFC hardware, this wasn't really an issue and still isn't for most people. However, either Google, LG, Samsung or all three decided to use another company's (Broadcom) hardware in the Nexus 4/10. While any NFC Tags made by any company that adhere to the NFC Forum's protocols will work fine, because the Mifare Classic does not meet those criteria it is not compatible with the Nexus 4/10.