The fact that Beowulf is fighting any of these foes is a testament to his fearlessness.  He chose to go defend Hrothgar against Grendel; because of that, he had to fight Grendel's mother; because of that, undoubtedly, he was named successor to King Hygelac and fought the battle against the dragon to protect his people.  That being said, there are some specific examples of fearlessness in Beowulf.

First, he chose to fight Grendel without weapons.  He made that choice because that's how Grendel fights; however, he was under no obligation to treat this enemy with honor, since Grendel was a marauder who had no intention of ever "fighting fair."

Second, he actually followed Grendel's mother to the bottom of the ocean to her lair.  She wasn't chasing him, nor was he in any danger from her.  Instead, Beowulf recognized the potential of another monster attacking Hrothgar's people and decided to preempt her attacks by killing her before she started.

Finally, Beowulf attacked the dragon in a single-handed combat.  This is an imposing fifty-foot-long, fire-breathing dragon who has been roused to anger.  Beowulf says,

...when he comes to me
I mean to stand, not run from his shooting
Flames, stand still till fate decides
Which of us wins.... (line 2525-2528)

That's what he says before the battle, and that's what he does.  He never backs away, even though his death is imminent.

These are three of Beowulf's biggest battles, and he fights each of his foes fearlessly.