What main peaks should be seen in the IR spectrum the methyl benzoate???

You should see the following peaks.
##”3100-3000 cm”^(-1) “(w)”color(white)(XXXXXXXXXXXXXl) “Aromatic C-H stretch”##
##”2960 and 2870 cm”^(-1) “(m)” color(white)(XXXXXXXXXXXl)”Methyl C-H stretch”##
##”1730-1715 cm”^(-1) “(s)” color(white)(XXXXXXXXXXXXXX)”Conjugated ester C=O stretch”##
##”1600-1585 cm”^(-1) “(m) and 1500-1400 cm”^-1 “(m)”color(white)(X)”Aromatic C-C stretch”##
##”1380 cm”^(-1) “(w) and 1260 cm”^(-1) “(m)” color(white)(XXXXXXX)”Methyl C-H bend”##
##”1300-1000 cm”^(-1)”(s) (three bands)”color(white)(XXXXXXX) “C-O stretch“##
##”750-700 cm”^(-1) “(s) and 710-690 cm”^(-1) “(m)” color(white)(XXXX)”Aromatic C-H ‘out-of-plane’ bend”##
The structure of methyl benzoate is
It has a monosubstituted benzene ring, an ester group, and a methyl group.
1. ##” C-O”## stretching
The strongest peak should be the ester ##”C=O”## stretch.
This normally appears at ##”1750-1735 cm”^-1##, but conjugation with the ring shifts the peak to ##”1730-1715 cm”^-1##.
There should also be one medium and two strong bands in the region from ##”1300-1000 cm”^-1##.
The two strong bands arise from the symmetric and antisymmetric ##”C-O”## stretches of the ester ##”COO”## group, and the weaker band is the ether ##”CO”## stretch of the ##”OCH”_3## group.
2. ##”C-C”## Stretching
The benzene ring has characteristic medium-strength ##”C-C”## stretching bands in the ##”1600-1585 cm”^(-1)## and ##”1500-1400 cm”^(-1)## regions.
3. ##”C-H”## bending
The monosubstituted phenyl group has characteristic strong bending vibrations at ##”750-700 cm”^(-1)## and ##”710-690 cm”^(-1)##.
The methyl group usually shows a weak band at ##”1380 cm”^(-1)## and a medium band at ##”1260 cm”^(-1)##.
4. ##”C-H”## Stretching
The aromatic ##”C-H”## stretch is usually weak and occurs at ##”3100-3000 cm”^-1##.
The aliphatic ##”CH”_3## group shows stronger symmetric and antisymmetric stretches at ##”2960 cm”^(-1)## and ##”2870 cm”^(-1)##.
Here’s the actual IR spectrum of methyl benzoate.

(from chemistrytextbookcrawl.blogspot.com)