- Multiaxis singularities: much less of a problem than it used to be. We don't need closed-form solutions any more; we have enough CPU power at the robot to deal with this. You need some additional constraint, like "minimize jerk" when you have too many degrees of freedom.
- Simultaneous Location and Mapping. SLAM for short: Getting much better. Things which explore and return a map are fairly common now. LIDAR helps. So does having a heading gyro with a low drift rate. Available commercially on vacuum cleaners.
- Lost Robot Problem: hard, but in practical situations, markers of some kind, visual or RF, help.
- Object manipulation and haptic feedback: Look up the DARPA manipulation challenge. Getting a key into a lock is still a hard problem. It's embarrassing how bad this is. Part of the problem is that force-sensing wrists are still far too expensive for no good reason. I once built one out of a 6DOF mouse, which is just a spring-loaded thing with optical sensors. Something I was fooling around with before TechShop went down. I have a little robot arm with an end wrench and a force sensing wrist. The idea was to get it to put the wrench around the bolt head by feel. Nice problem because a 6DOF force sensor gives you all the info you can get while holding an end wrench.
- Depth estimation: LIDAR helps. The second Kinect was a boon to robotics. Low-cost 3D LIDAR units are still rare. Amusingly, depth from movement progressed because of the desire to make 3D movies from 2D movies. (Are there still 3D movies being made?)
- Position estimation of moving objects: the military spends a lot of time on this, but doesn't publish much. Behind the staring sensor of a modern all-aspect air to air missile is - something that solves that problem.
- Affordance discovery: Very little progress.
The real problem: solve any of these problems, make very little money. If you're qualified to work on any of these problems, you can go to Google, Apple, etc. and make a lot of money solving easier problems.