Stop the Singing
I have a coworker who spends all day humming and sometimes singing to herself, and looking up music videos online. It is often distracting to the point where I need to get up and work somewhere else for a while. Should I say something to her? Does it make a difference that she is singing hymns and religious songs about Jesus, and looking up performances of similar songs? Does it make a difference that she's the most senior member of our team both in terms of age and experience or that she was the one who trained me? As the only Jew in my workplace, and because of the other issues mentioned above, I want to tread carefully. I don't want to infringe on her religious liberty, but I also want to stand up for myself and create a more pleasant working environment. She tends to make fun of her voice and say she's a horrible singer (which unfortunately is true), but I don't want to go that route.
Stop the Singing
If your life were a movie, there would be a scene where you come into work and play the most recent Maccabeats video for your co-workers, and everyone would start singing and dancing together. It would be a beautiful, heart-warming, cross-cultural experience. But your life, as far as I know, doesn't have that level of production value, and if you come into work and suddenly start blasting Hatikvah, it's just going to create more noise and more distractions and single you out even more as the only Jew.
While this kind of work situation can make life really unpleasant, and while you're right that you have to tread carefully, that doesn't mean you can't tread at all. In the long run, it's definitely worth taking some potentially uncomfortable steps in order to create a more pleasant working experience for yourself. So apart from creating a big Hollywood moment, there are two ways to go with this: avoidance and confrontation.
Let's start with avoidance. Since tuning it out isn't an option, can you block it out? Bring your own headphones to work and listen to something that you like. Download a white noise app if you have to, but at least you won't hear her singing. You say that sometimes you physically move to another location, so you could look into whether that's a more permanent option. You could also set yourself up for the day at the other location so you don't wait to move until you're already annoyed.
Since this person trained you and you work closely together, she might sense that something is wrong if she doesn't see you at your normal work space, and you still might have to explain yourself, in which case, you could decide to head right to the confrontation option. Tell her that you're having trouble focusing, and you'd really appreciate it if she could listen to the videos through headphones and try not to sing. Say that you know it's a habit and probably helps her pass the time, but you know you're not being the best employee you could be because you're wasting valuable energy trying to concentrate over the music. Hopefully that will resonate with her as your higher-up, and she'll want to help you maximize your performance (at your job, not as a back-up singer). As tempting as it is, don't mention her terrible voice or the religious nature of the songs. Bringing up either of those issues will unnecessarily complicate your ultimate goal, which is for her to stop singing.
Spending any more time fuming without taking action really will start to affect your work and your relationship with your co-worker. Standing up for yourself about this will make other workplace issues easier to address when they come up. Plus, if you ever do make a movie of your life, you'll get to portray your character as totally assertive, respectful and professional, all of which are more valuable than a production number.