To build any scale, its best to create the succession of steps and half-steps typical of that chromatic scale, starting from the note that will be the root in the new scale. It's obvious that the notes will change with respect to the root selected.
Lets try to build the MDS of E. First of all, we make reference to the chromatic scale, starting the succession of notes with the E note.

Now we use the succession of steps and half-steps typical of the MDS, and we insert it to start with the E. The result is the following:

At this point we put the notes of the MDS in E in succession.

You can clearly see the presence of the alterations on the F, G, C and D notes. These are typical for the scale of E major and, for convenience, it would be best to identify them in key. This system, that we will analyze using the relationships of the scales, is called armature in key.

Being diatonic, every note of the MDS must have a different name. If we look at the MDS of F, we see that the A# note has been transformed into Bb: this is obligatory, because in the construction after the A note we have to add a half-step (getting A#), but we would also have found two consecutive A's and no B (A - A# - C). The substitution of A# with Bb is therefore obligatory. This is the MDS of F, with 1 flat in key (Bb).

In some scales we can find the sharp on the E and B notes or the flat on the C and F. For example in the MDS of Gb we find the flat on the C note, as we can see below.

When building a scale its possible that, in some scales with many alterations, the sound of a note coincides with the following note. We have seen that in the scales, thanks to the rule which indicates if it is diatonic (every note must have a different name), that all the notes are present in the succession of the sounds. This can involve the presence of the double sharp (which is written as ##, or with the symbol x), or the presence of the double flat (two bb's before the note).
As an example we will build the major diatonic scale of A#, remembering to visualize the chromatic scale and to perform the various inharmonic transformations.
We will add a tone to the root, which is A#. We immediately find ourselves at the first inharmonic transformation, because the C note must be a sharp, that coincides with the sound of the C. And since we use the name C on this note we skipped the B, this is why we must use the B#.
From the B# we advance a tone, coming to the double sharp (C## or Cx). This note has the same sound as the natural D note, but in the scale it assumes the name of C##.
From C## we advance a semi-tone (half-step) and we find the D#. You can use common sense to compare the C## to the D (its inharmonic note) and thinking if you add 1st to the D this jumps to D#. Therefore C## (which is D) plus 1 half-step becomes D#.
From D# we decrease a tone and find another inharmonic sound. In fact we need to call it E# (coincides with the sound of F).
From E# we decrease a tone to F## (which has the actual sound of G natural).
From F## we come to G## (actual sound: A natural), to conclude 1 half-step above the root of A#. Look at the scale in both directions (ascending and descending), remembering that the x identifies a double sharp:

The same is also true for all types of scales. You only need to learn rules that govern their construction. Look at the various examples of the minor scales and you will see how many possible alterations there are.
It's very useful to know all the scales, their construction, tonality and type (major, minor, modal, etc.) and to recognize the scale from the alterations in key (or armature in key). Some good advice is to practice to build the scales, meticulously following the rules. You can practice by comparing the results to the table of the Major Diatonic Scales.
I want to mention a very important detail. When you write a musical piece that has momentary alterations (those not found in key), the rule of the alterations would force us to ascend to identify the sharps and descend to identify the flats. Instead, it's much easier to write the piece with the least possible amount of symbols to facilitate the reading. Look at the following line.

Notice that in the last two octaves of the first bar there are two descending notes, and therefore the last F# must be written as Gb (according to the rule). Leaving the F note (which automatically is sharp because of the alteration on the preceding F#), I avoided to add another symbol (the flat on the descending G). The same in the last two octaves of the second beat, where I preferred to leave the F# in place of the Gb. This is all to facilitate the reading.
Many times there are scale changes within the song. These can be identified by the insertion of a key followed by the new alterations at the point where the scales change. But it is also common not to find alterations in key, even at the beginning of the song. This normally happens when the melodic line has many alterations, or when the scale changes are very close to each other.