I realize this first piece of advice won’t be of use to many people reading, but I thought it was worth mentioning. Just as is the case when applying for full-time jobs, your best bet for landing an interview, and potentially securing the position, is to have somebody who knows the company, someone at the company or even the person doing the hiring suggest you.
That will ring true for an internship and for a full-time paid job, though you don’t need as bright of a recommendation for an internship. Most unpaid positions (or low paying positions, as they are typically reserved for young people) aren’t high-stakes, so don’t worry too much if you don’t have an “in,” as they say.
Now, this isn’t always possible, but since it’s important, start thinking about who you know, and then, who they know. Maybe one of your Professors has a connection to a musical company or somebody involved in the industry who could use an Intern. Perhaps the person who has been giving you lessons or teaching you how to sing can help write a recommendation, or, if you’re really lucky, you already have a family member or family friend who knows someone who will be able to help you.
These connections are difficult to find at first, but as you begin volunteering, working, playing, auditioning, and interning, you’ll meet more people, and all of a sudden your network will grow considerably, and that will be handy when you are looking for your next internship (and eventually, job).