Assuming the character is using the scroll by themselves, does the character need both hands available to unroll a scroll; or only one free hand to hold the scroll and manage the unrolling while the other hand can be be holding an object but still be used prevent the edge from rolling back over the text? 

I believe you only need a single hand. 
And a scroll doesn’t have to be a rolled up scroll, per se. It could easily be a sheet of paper. A rock. A wax tablet. What it’s made of is pure flavor. The important parts are it’s on your spell list and it’s single use. 

How many hands is required to read a Spell Scroll is not strictly defined in the rules and as Xalthu points out could take many different possible forms.

Generally it is assumed to take at least one hand, but I have seen posts talk about glueing a Spell Scroll to the inside of a shield. Thus the scroll can be held up and read by the same hand that donned the shield.

One thing to be aware of if you want to be particularly pedantic about the rules is that a Spell Scroll only provides the material component of the spell (if it has one). So a spell that requires a somatic component would still require a free hand to perform it when cast from a Spell Scroll.

Quote from petepan3 >>

Assuming the character is using the scroll by themselves, does the character need both hands available to unroll a scroll; or only one free hand to hold the scroll and manage the unrolling while the other hand can be be holding an object but still be used prevent the edge from rolling back over the text? 

 A Spell Scroll  doesn't require free hand per se to be read. Some magic items must be held to be used so depending on the scroll fabric or shape it would be up to DM wether one require 1-2 hands or if it can be read without being held. A character at least typically use a free hand to pull it out anyway.

Wearing and Welding Items: Using a magic item's properties might mean wearing or wielding it. A magic item meant to be worn must be donned in the intended fashion: boots go on the feet, gloves on the hands, hats and helmets on the head, and rings on the finger. Magic armor must be donned, a shield strapped to the arm, a cloak fastened about the shoulders. A weapon must be held.

Spell Scroll: A spell scroll bears the words of a single spell, written in a mystical cipher. If the spell is on your class’s spell list, you can read the scroll and cast its spell without providing any material components. Otherwise, the scroll is unintelligible. Casting the spell by reading the scroll requires the spell’s normal casting time. 

Scroll: Whatever the nature of the magic contained in a scroll, unleashing that magic requires the user to read the scroll. 

Quote from Fangeye >>

One thing to be aware of if you want to be particularly pedantic about the rules is that a Spell Scroll only provides the material component of the spell (if it has one). So a spell that requires a somatic component would still require a free hand to perform it when cast from a Spell Scroll.

One quick comment on this.  There is a general rule about magic items and a specific rule about the spell scroll which addresses this:

Spells

Some magic items allow the user to cast a spell from the item. The spell is cast at the lowest possible spell and caster level, doesn't expend any of the user's spell slots, and requires no components, unless the item's description says otherwise.

A spell scroll bears the words of a single spell, written in a mystical cipher. If the spell is on your class’s spell list, you can read the scroll and cast its spell without providing any material components.

So, is this the item description saying otherwise, or not?  Does the fact that you explicitly do not have to provide material components actually imply that you DO need to provide verbal and somatic components?  Enough to override the general rule that these are not required?  Either way, that could have been spelled out more clearly imo.

While not official ruling, the Dev answered this on Twitter before;

@DakotaHansen13 Casting from a spell scroll. It doesn't say it requires V/S components, but specifically lists not using M. Does that line override the general rule about not needing V/S/M for any spells from magic items, so you would need V/S?

@JeremyECrawford Spell scrolls follow the normal rule for casting a spell from a magic item: you don't need to provide any components to cast the spell (V, S, or M). Spell scrolls have a twist, though: you must read the scroll to cast its spell. This is effectively an ad hoc component.

Thank you all for the responses.  Great discussions and very helpful. Cheers!