The inverse Galois problem, what is it good for? Several years ago I attended a colloquium talk of an expert in Galois theory. He motivated some of his work on its relation with the inverse Galois problem. During the talk, a guy from the audience asked: "why should I, as a number theorist, should care about the inverse Galois problem?" 
I must say that as a young graduate student that works on Galois theory, I was amazed or even perhaps shocked from this question. But later, I realized that I should have asked myself this question long ago. 
Can you pose reasons to convince a mathematician (not just number theorist) of the importance of the inverse Galois problem? Or maybe why it's unimportant if you want to ruin the party ;)  
 A: For me, it's one of those questions that would not be so interesting if the answer is Yes but which would probably be very interesting if the answer is No. If not all groups are Galois groups over Q, then there is probably some structure that can be regarded as an obstruction, and then this structure would probably be essential to know about. For instance, not all groups are Galois groups over local fields -- they have to be solvable. This is by basic properties of the higher ramification filtration, which is, surprise, essential to know about if you want to understand local fields. So you could say it's an approach to finding deeper structure in the absolute Galois group. Why not just do that directly? The problem with directly looking for structure is that it's not a yes/no question, and so sometimes you lose track of what exactly you're doing (although in new and fertile subjects often you don't). So the inverse Galois problem has the advantage of being a yes/no question and the advantage that things would be really interesting if the answer is No. Unfortunately, I think the answer is expected to be Yes, though correct me if I'm wrong.
A: Any branch of mathematics after the first few definitions will make everyone routinely ask themselves some basic questions. I consider  Inverse Galois Problem is one such. If the question (i) is not highly technical, (ii) can be understood at very early stages and (iii) does not sound concocted then it justifies itself. 
These are the natural questions the subject should attempt to answer. (It is irrelevant if solving them requires Fields medallists  or undergraduates).
Let me list more questions in the same category (not necessarily of the same level of difficulty!)


*

*Which divisors of $|G|$ are orders of subgroups of $G$?

*Which  connected open subsets of the complex plane are biholomorphic to the unit disc? 

*For which numbers $d$, is the ring  $\mathbf{Z}[\sqrt d]$  a UFD? 

*Which finite groups occur as subgroups of $\mathbf{SO}(3)$? 

*Which integers are represented by an indefinite/definite integral quadratic form?

*Which projective curves are subvarieties of the projective plane?
I have been under the impression that this is  the way mathematicians think.
 If someone  questions the relevance of the above questions it would be difficult for me to communicate with that person.
A: The question is not really my business, but I can give a stock answer.  It might be a good answer in the sense that it convinces an outsider like me.
The narrowest version of the inverse Galois problem, find all of the Galois groups of finite extensions of $\mathbb{Q}$, might not be all that interesting.  A better question would be the following:  Let $G$ be a finite group and let $\mathbb{F}$ be a field of characteristic 0 (or more generally a perfect field).  Can you describe the set (or moduli space if you like) of all Galois extensions of $\mathbb{F}$ over $G$?  For instance if $G = C_2$, it's a good question with a good answer; the question is a model of taking square roots of elements of $\mathbb{F}$.  With that special case in mind, it's always a good question.  It can be viewed as a theory of nonabelian surds.
If for a given field $\mathbb{F}$ and a given finite group $G$, you don't even know if there are any points in the moduli space of extensions with Galois group $G$, then you hardly know anything.  In particular, $\mathbb{Q}$ is an important field, and there are many specific finite groups for which people don't even know that much.
A: The previous answers are all on point; let me just say a little more.
First, the inverse Galois problem (IGP) as a problem is a sink, not a source (or something with both inward and outward flow!): I know of no nontrivial consequences of assuming that every finite group over ${\bf Q}$ (or even over every Hilbertian field) is a Galois group.  This does not mean it's a bad problem: the same holds for Fermat's Last Theorem (FLT).
As with FLT, if IGP were easy to prove, then it would be of little interest. As a good example, if you know Dirichlet's theorem on primes in arithmetic progressions, it's easy to prove that every finite abelian group occurs as a Galois group of ${\bf Q}$. What does this tell you about the maximal abelian extension of ${\bf Q}$? Not much. The Kronecker-Weber theorem is an order of magnitude deeper.  But as with FLT, the special cases of IGP that have been established use a wide array of fascinating techniques and provide an important border-crossing between algebra and geometry.  
Arguably more interesting than IGP itself is the Regular Inverse Galois Problem (RIGP): 

For any field $K$ and any finite group $G$, there exists a regular function field $K(C)/K(t)$ with Galois group isomorphic to $G$.  

If $K$ is Hilbertian (e.g. a global field) then RIGP for $K$ implies IGP for $K$.  Now RIGP is of great interest in arithmetic geometry: given any finite group $G$ there are infinitely many moduli spaces (Hurwitz spaces) attached to the problem of realizing $G$ regularly over $K$ (because we have discrete invariants which can take infinitely many possible values, like the number of branch points). If even one of these Hurwitz schemes has a $K$-rational point, then $G$ occurs regularly over $K$. In general, the prevailing wisdom about varieties over fields like ${\bf Q}$ is that they should have very few rational points other than the ones that stare you in the face. (Yes, it is difficult or impossible to formalize this precisely.) So it is somewhat reasonable to say that the chance that a given Hurwitz space, say of general type, has a ${\bf Q}$-rational point is zero, but what about the chance that at least one of infinitely many Hurwitz spaces, related to each other by various functorialities, has a ${\bf Q}$-rational point?  To me that is one of mathematics' most fascinating questions: to learn the answer either way would be tremendously exciting.
A: Just to illustrate that the inverse Galois problem in specific cases can be useful.
In my master thesis (2016-2017), I used the inverse Galois problem of cyclic groups over $\mathbb{Q}$ to construct Anosov $2$-step nilpotent rational Lie algebras of certain types. This can help in the classification of Anosov nilmanifolds.
A: I personally know of no immediate applications of a positive (or negative) answer to the inverse Galois problem. At the same time, the problem seems to me a useful standard against which to gauge mathematical progress. 
Answering the inverse Galois problem for solvable extensions required class field theory (one of the pinnacles of early 20th century mathematics). This can be seen as evidence that the ability to solve the inverse Galois problem will entail a deeper understanding of a variety of mathematical things.
A: just a link, 
Families of number fields of prime discriminant
if we not only consider about the group, but also put some other restrictions on the extension (such as discriminant, ramifications etc), then we have the above problem, which is quite interesting.
A: Bauer's Theorem (a simple consequence of the Chebotarev Density Theorem) states that a finite
Galois extension K of an algebraic number field F is uniquely determined (as a subield of
some fixed algebraic closure of F) by the set of primes of F which split completely in K.  Thus knowing all possible Galois groups is the same as knowing all possible splitting laws
in finite Galois extensions.  Being able to describe these splitting laws in some explicit
fashion is basically "nonabelian reciprocity", which is THE most important problem in algebraic number theory, so the "inverse Galois problem" is of FUNDAMENTAL importance to all
number theorists.
