I am frequently hearing of new research finding how bad situps and crunches are (example). So what is a safe way to exercise the abdominal muscles? For example is doing the plank proven to be a safe exercise?
Abs can be sufficiently trained via isometric exercise. I believe the safest way to exercise the abdominals is to use them for their intended function (stabilization) under a progressively increasing load or difficulty.
Besides the exercises recommended in the article you cite (planks, bridges, leg-lifts, bird-dogs and "stirring the pot"), full-body exercises that rely on your core muscles are also great ways to improve abdominal fitness, and they often strengthen your back at the same time, thereby helping prevent injury instead of potentially causing it.
Some examples of those are swimming (especially dolphin, butterfly and other strokes that rely on the abdomen -- dolphin-kicking with fins on is one of my favorite ab workouts, possibly just because it's so fun), tai chi (choose a style appropriate to your fitness level), surfing (or skateboarding, or slacklining; anything where you have to balance your body) and rock-climbing (if done properly, focusing on balance and technique rather than just arm-strength). You'll be hard-pressed to find anyone who's a regular at any of those who doesn't have a nice strong stomach.
If you're just looking for one exercise to add to a workout, I'd go with planks. The plank and exercises built on it are incredible stomach-strengtheners -- we use those on my swim-team as daily ab-exercises and they're intense. As with all exercises, form is crucial if you want to avoid injury, so double-check and make sure you're doing it correctly.