"Susan, although very short, found herself reaching for the bottle of soda on the top shelf."
I burned my arm on the autoclave reaching for a speculum.
Yes, "reaching new heights" is a phrase that can function as a sentence fragment, but it is not a complete sentence on its own. For it to be a complete sentence, it would need a subject and a verb, such as "We are reaching new heights." As it stands, it conveys an idea but lacks the necessary grammatical structure to be a full sentence.
Ellie lived in the garret of the house, she had to walk many stairs before reaching her small loft.
Once the river slowed on reaching the valley bottom, it began to meander the short distance to reach the sea.
The river makes many sinuous turns before reaching the delta. The sinuous moves of the cobra do not confuse the mongoose.
The spindly tree stood tall in the forest, its thin branches reaching out in all directions.
Her fingers stretched out to the sky as if reaching for the clouds.
His singing was so high, it must have been in falsetto.
Control your velocity, or pay a one dollar fine.
Determination was one of my ways reaching my goal.
After the flood,concerned citizens made benevolent contributions for the relief effort.
What do you mean by a line? If you mean a sentence, then yes, just like this. If you mean a natural line break that results from a sentence reaching the end of the paper, then yes, just like that. If you mean an artificial line break like this that does not come from the sentence reaching the end of the paper, no, you can't do that.