You can try to soften the spine by opening it in various locations and flexing the book gently.
(Here's a nice version of the procedure , with diagrams, even; and here's another , similar graphic, this one from a bookbinder.)
If that doesn't work, you can get a piece of clear plastic and leave it over the pages; not an ideal solution, but it will let you use the book.
In the end, don't be too concerned with keeping the book in mint condition, unless this is a valuable or rare volume.
Break the spine if need be; you can always repair the book later, or hold it together with a rubber band.
A book of scores is of no use if you can't use it.
music.stackexchange.com/users/59/Goodbye Stack Exchange
A great longterm fix that I've found for this issue is to just take the score to a kinko's or other copy center and have them replace the binding with a spiral binding.
This way, the book can stay open on any page and you don't have to worry about the binding breaking.
The only thing to watch out for is to make sure you don't lose any of the music when you have them change the binding over.
But other than that, it's a great $5 investment!
music.stackexchange.com/users/1546/Reina Abolofia
Book spines are intended to keep the book in its closed shape.
As they age and are repeatedly opened, they start to stay open because the spine gets damaged.
If you're careful, bending and folding won't make it any worse than the "natural" damage over time.
Another option is to cut out the spine altogether.
You could punch holes in the margins and use a binder or something to keep everything together.
music.stackexchange.com/users/None/
I use clothespins to keep the pages open. :D
music.stackexchange.com/users/120/Raskolnikov
When I was growing up, my dad came up with a brilliant strategy for a stubborn binding: Bandsaw off the spine, then spiral-bind the whole book.
Voilà!
Well-behaved Mussorgsky.
music.stackexchange.com/users/1212/buildsucceeded
The "Standard Method" for getting a book to stay open is to Place the spine firmly on a table.
open both covers and fold flat
(run your finger along the inside to help guide the fold on paperbacks) fold down each page in turn, alernating sides, until you get to the middle.
This way should leave the spine intact, but able to lay flat.
music.stackexchange.com/users/1344/luser droog
See my answer to another question Copy the sheet music to separate sheets and lay them out side by side.
If they don't fit, memorize the easy pages until you can discard them.
Eventually memorize the whole song and become more impressive to watch.
(this one can wait)
I know some people have a hard time with the memorize part, but I have gotten good at it
and I hate page flipping :)
music.stackexchange.com/users/86/musicwithoutpaper
With some score books, if you try to open it and put it on a support, it will close itself.  For the moment I have to force and fold it but it damages the spine of the book.  Do you have techniques to open those books without damaging them too much?
How to handle a newly purchased score book that won't stay opened?
music.stackexchange.com
<sheet-music><books>
music.stackexchange.com/questions/188
music.stackexchange.com/users/141/Benoit
To soften a newly purchased score book you can try to soften the spine by opening it in various locations and flexing the book gently. Use clothespins to keep the pages open. Break the spine or have it replaced with spiral binding.  To keep the pages of a new book open you can try softening the spine, but careful bending, or use pegs or a piece of plastic to keep the book open. Alternatively you could break the spine and spiral bind the book
Soften the spine, by careful bending and folding.  Keep the pages open but using clothes pins/pegs or a piece of plastic  Break the spine and spiral bind the book