A quick search on your phone and you will quickly find that, yep, there certainly are a TON of running apps to choose from. Some are apps that are more focused on the broader scope of health and fitness, and include not just the ability to track running metrics but can keep up with metrics needed for tracking other methods of exercise. Still others are dedicated to running, but throw in there ways to track other important factors that play into one's running performance, such as tracking one's sleep and recovery cycles, tracking one's nutrition and the caloric and macronutrient breakdown of each snack and meal, and reminders to get enough water in throughout the day to stay hydrated.
One of the very first running apps on the scene, and one of the most popular even still, is the MapMyRun App. Runners like it because it keeps track of time and distance when you are outside running the roads and trails. It is great if you are running new-to-you trails and do not know how far they go. It is also a free app, and you can have it on tracking your distance while you are still listening to music from your phone. As far as accuracy is concerned, MayMyRun is very accurate - so long as the GPS locator on your phone is accurate. And this is true for most running apps. The apps themselves are fairly reliable when it comes to GPS signal and distance tracking, but it is often other factors such as the phone itself or the weather outside (i.e. cloud coverage that might block a satelite signal) that inhibits accuracy.
Another great running app is Runkeeper, but this one only has limited ability in the free version. Splurge and pay for the full access app because runners are raving about the accuracy of Runkeeper in tracking distance, pace, and mile split times. This app also syncs with your phone and offers you guidance along the way, even while continuing to play your music (i.e. super helpful for not getting lost!)
If you are still concerned with accuracy, you can always invest in a GPS watch as well. These can be expensive, but if you are serious about getting serious with your running, and intend to continue tracking your runs, then a GPS watch would be a wise investment. Check out our list of great GPS watches here!