Some of these songs were already a hit on their own, but I tend to associate them with the following movies they were featured in. To limit the many options of iconic songs used throughout the 90's, this list of 10 items were songs made by bands. The movies also happen to come from the mid to late 90's.
This post was inspired by the good sis 
tres_francais with the Olsen twins' birthday post, where I was remembering how much I loved Passport to Paris!
10. Sixpence None the Richer - "Kiss Me"
🎥 She's All That (1999)
I just remembered how Freddie Prinze Jr. was THE teen heartthrob of the 90's *ugh yas godt*. "Kiss Me" appeared when miss Rachael Leigh Cook stepped out after her makeover for her big reveal, and is the movie's main song.
9. The La's - "There She Goes"
🎥 The Parent Trap (1998)
This song was featured when Hallie finally landed in London. The Parent Trap soundtrack featured a lot of covers, and coincidentally Sixpence None the Richer made a cover of "There She Goes" the year after.
8. The Verve - "Bittersweet Symphony"
🎥 Cruel Intentions (1999)
"Bittersweet Symphony" was used at the end credits of Cruel Intentions. The Verve split up during the height of their success when this song was breaking.
7. Stretch Princess - "Sugar"
🎥 Passport to Paris (1999)
The song that inspired this list, and was used during the montage of Mary Kate and Ashley with the French model trying on clothes at different stores.
6. Garbage - "#1 Crush"
🎥 Romeo + Juliet (1996)
This movie adaption was bizarre, but maybe it was tooaheadofitstimeforsomepeople.gif. This was my introduction to Garbage.
5. Wang Chung - "Dance Hall Days"
🎥 Romy and Michele's High School Reunion (1997)
An epic soundtrack and a lot of stand outs used for the movie. I chose "Dance Hall Days" cause Romy was gettin ha life and whippin that pony.
4. Mono - "Life In Mono"
🎥 Great Expectations (1998)
OP has not seen this movie adaption of Great Expectations. Don't know much about this band but this is a beautiful song and the band's most well known song.
3. General Public - "Tenderness"
🎥 Clueless (1995)
The Clueless soundtrack was pretty epic too. "We're Alright" by Supergrass could be considered the main song of the movie, but "Tenderness" gave us a schmaltzy happy ending and a wedding scene.
2. Rusted Root - "Send Me On My Way"
🎥 Matilda (1996)
Miss Matilda was giving the girls a good #read and serving (pancakes) for the gawds with this song used during the pancake making montage while she was home alone, as well as the ending when she was moving in with Miss Honey. Nostalgia overload!
1. Texas - "Put Your Arms Around Me"
🎥 Ever After (1998)
This song was featured at the very end and clashed with the 16th century feel of the movie. An ethereal bop.
HOOOOO HoooooOOOOOOOO
so put your AAAARMS around meeeEEEEe
You LET me believe that you are someone else
HOOO Hoooo HOO
Cause only tiiIIIime can take youuuuu
So let me believe
That I am somewhere eeeeeeeeeeeelse
Sources: 12345678910