There are certainly many inequalities which are rather important and useful and which appear frequently in various areas in mathematics (AM-GM, Jensen, Cauchy-Schwarz, etc.) The question I want to ask is whether some of them would be also useful as tags on this site. And if yes, for which of them tags should be created.
To avoid making this question too general, let's deal just with the inequalities concerning real numbers. So for the purpose of this question let us leave inequalities from probability (such as Chebyshev's inequality), analysis (Harnack's inequality, Grönwall's inequality, etc.) and various other areas aside.
If we decide that some the inequalities should have their own tags, we should also discuss when these tags are supposed to be used. Are they only for the questions about these inequalities? Should they be added to the questions where the OP explicitly says that they want to see solution using this particular inequality? Or if some an answer is given which uses some inequality, should the tag be added based on the answer?
One more thing to keep in mind is that question can have at most five tags. So if we create too many very specific tags, it might happen that on some question we will not have enough space to add all tags which might be suitable there.
On the other hand, I can see that inequalities tag might start to resemble a big monolith of questions which are not divided into subcategories, which make questions about inequalities rather difficult to search. (Although I am not sure to which extent they can be reasonably categorized. And whether tags could actually improve searching among these question, or whether it is simpler to search for the exact formula.) It would be great if some of the users who are active in the inequality tag could comment on whether they think that inequalities tag needs to become more organized.
The reason why I have decided to post the question now was that not so long ago two tags of this nature have been created. Namely a.m.-g.m.-inequality was created in January and young-inequality created very recently. (Another similar tag which existed, although only for a very brief period, was cauchy-schwarz.)
In both cases the tags start growing relatively fast. Therefore I considered asking about community opinion as a reasonable thing to do. If the community consensus is that those tags should be removed, it is better to find out before the tags contain too many questions and removing the tags is a lot of work. (Although moderators can remove tags without bumping - see here and here - but this only works if there is another tag where all currently tagged question would fit. Possibly in the case of these tags, inequalities might be a good fit for most questions, so it is possible that in this case we will not have to do manual retagging of all questions if the tags are removed.)
EDIT: Recently (end of July 2017) two new inequality-related tags were created, namely cauchy-schwarz-inequality and holder-inequality. The list of questions currently having these tags can be found here.
EDIT3: The tags karamata-inequality and tangent-line-method were added today. The latter is a method for proving inequalities rather than a specific inequality - the same is true about the tags mentioned in Jyrki's answer. This kind of moved the question a bit into contest-math.
I guess the above edits show that number of inequality-related tags is still growing - so I will not post further updates when new such tags appear. (I guess the above are sufficient as examples.)