Introduction
As guitarists, our exercises are filled with all kinds of diatonic and chromatic scales or chordscales, along with various techniques, licks, and comping methods. Today, I’d like to present a very efficient and effective method of practicing scales, and that is the Single-String Scales. This is a great way to getting familiar with the fretboard scale-wise, breaking out of the old habit of staying within a single position, and thinking more musically. This exercise has been mentioned in books such as The Advancing Guitarist and The Serious Guitarist - Right Hand Picking.
Single-String Scales
So how does Single-String Scales (SSS) work? There are three levels of limitations that you can apply when practicing.
Level 1:
Imagine that your guitar only has one string - any string - and play a scale with the note of the open string as the tonic, e.g., play an E scale over the e string.
If you are having difficulties deciding what scale type to play with, try the major scale. Since all scale degrees are based upon the major scale, it never hurts to be more familiar with them. As a hint, here are the fret positions of the major scale degrees: 
Memorize them as if your life depends upon it, and figure out the positions above the twelfth fret on your own.
Try to practice all intervals, play with any sequence you can think of, and alternate between both legato and staccato playing styles. In essence, make music with it. I will provide some ideas for possible exercises listed below:
Interval of thirds:  
Sequence of three notes: 
Sequence of two notes with open string in between: 
Sequence of three notes with open string in between: 
Once you got the major scales under your fingers, start experimenting the same idea with different scales and modes, and see what you can come up with. Often times, the melodies we create with this method can result in a fresher sound compared to playing with fixed positions that some of us extensively rely upon.
Level 2:
After you become familiar with playing a scale with tonic on the open string, do the same thing with tonic on the first fret. This time, you might not be able to take advantage of the open string when changing hand positions, and thus requires cleaner and more careful movements for that purpose. Expect wider stretches and jumps in general. 
When you got that in your pocket, try setting the tonic on any fret so you will gain complete control and freedom over any scale in any position - all on one string. You may need to focus on the note names when making a wide positional jump. That’s a musical sign.
Level 3:
This time, combine adjacent strings. Since now you have no problem playing on the e string and B string by themselves, or any other pair of strings, combine them. Play melodies on two strings. You will have easier time with arpeggios and more freedom experimenting wider melodic intervals.
Tips and Suggestions
When you are familiar with the system, try using only one finger to play every note for each run while staying connected or legato without the portamento effect (sliding), to improve your speed and precision. Also watch out for your right hand, the string is picked/plucked during left hand finger movement, you will not have a pretty tone.
Use different combination of fingers.
Try different fingerings when playing the scale.
After you have “unlocked” level 3, begin experimenting with double stops or more combination of notes.
Remember to consciously avoid falling back to your old positional habit if you are still holding onto them, and begin adding more strings into your selection.
Conclusion
As you reach the point of having all six strings as your resource, in fact even during your level three practices, you can take advantage of another technique called 3-Note per String, another great way to play with scales especially for faster passages. I will explain that in greater details in later posts.