So before I begin I want to saying I believe that losing ones job isn’t the worst thing that can happen most of the time. If your financially stable, losing ones job isn’t the worst thing that can happen, being bored is the worst thing that can happen. If your bored, your wasting your time, and lets be real you can’t buy more time. Challenging the status quo in any system carries some risk, but I believe most people can take on more risk than they usually do, but don’t due to fear of losing their job.
So, there’s a few ways to go about trying to “get better work”. The simplest is to ask for it, and keep asking for it, and until you get a hard answer (yes or no), you keep asking. Ask anyone for anything related to what you want to do. Your goal is to make it visible you want change for yourself. Without making this visible nothing will ever happen.
Odds are you wont be qualified to do most of the work you want, so you will have to do some homework, and get those skills required. For example, if you want more development work, or want to use the newest latest and greatest framework, you should have at least some experience in it. You can get this experience by doing some side projects by yourself. Unless your org provides resources, your probably on your own. If you make it visible you know some X, because your interested in it then its more likely you might get a chance to work on something using X. Without taking the initiative, odds are nothing will change.
Now if the changes you want required organizational changes, things are tougher to change. Not many organizations are flexible enough to allow for an individual to change where they are “placed within the machine”. The most important thing is making it visible you want the change, and take the steps necessary to start moving in that direction.
I personally went from IT work to more development work because I jumped heavily into any opportunity I got for development work. I went out of my way to find it, do it, and invested a lot of time and energy to just getting it, and obviously doing it. I asked at least 3 separate managers outside of “chain of command” if they had any development work available, and asked a few co-workers to automate some of their daily tasks. During this time I still performed my normal duties, but focused on any opportunities for dev work. Eventually all three managers found development related work (largely related to automation), which eventually lead to a real dev project down the line.
My situation didn’t seem as strict as yours, and I have limited experience with large organizations, so it might be more difficult depending on how the culture is. I just believe the main thing to focus on is making it visible and know what you want to do. What you want are managers who find the work for you. If management doesn’t know of what you want, or how badly you want to do it, then it will never come.
I want to point out that in some situations the work just might not be possible without drastic organizational changes. For example if most software development is done via contractors, and or offshore developers, getting anywhere into this process is somewhat a mess. Dealing with is very situational, so its tough to give advice for these sorts of situations.
Finally, there is always the obvious choice of moving on to another job/role entirely. Some companies do provide routes for people to change positions, since on-boarding processes can be simplified, its easier for companies to “move” an employee than to hire a new one. There’s also just finding a new job, but obviously this is its own game 
Good luck, keep bugging people and building things 