As you would expect a important civic institution built at the start of the 20th century has some carvings on it reflecting its purpose. The back (quick digression, there is a urban legend that the building is the wrong way around. This is probably not true) is where you find them wrapped around the entrance.
The area above the arch has a bunch of women blowing horns and plying cymbals in the middle there is a small bust of a bishop blessing you. I am not sure what the significance of this is unless they are playing you in with a fanfare and the bishop is blessing you for using the museum. Below the arch are three figures another bishop and two female figures. I think the Bishops are St Mungo representing Glasgow. The two female figures you can see below. One appears to be reading a book and the other has a small organ so I would guess they are art and learning.
The music one is actually quite creepy.
Finally we have the two sides.
This is the left side
The odd looking central figure is mercury since he has a winged hat and a serpent staff.
On the left there are several figures the one furthest left as you look at the picture holds what could be some sort of tool (Even zooming into it in the original photo I cant make out what it is) The next figure has a suspension bridge. On the right we have figures holding a bottle, a scroll and what I think is supposed to be a loom. This all indicates Science, Engineering and manufacture reflecting the practical scientific part of the museum.
This is the right side
Its not obvious who the central figure is this time though they appear to have a scroll across their knees. To the left this time is a figure with a small organ and two people with pan pipes the others appear to be singing. On the right are more singers there is also what looks like a small statue being carried by one of the figures. This is the side of the arts.
If you take this as a whole then its telling you that Glasgow represented by St Mungo is blessing you with a museum showing you education, Science, Engineering, manufacturing Art and Music.