Edited: There is a better way to do this mod and it should be done this way instead of downloading my mod. It can be done by editing the database.bin file using the Ryder Database Editor. Apologies for not knowing this before, I just found out about it today.
So here's how to do it yourself (if you don't already know about it):
1. Backup your Steam\steamapps\common\f1 2012\database\database.bin
2. Open the Steam\steamapps\common\f1 2012\database\database.bin file
3. Open the track_model and tyres tables
4. Note the tyres IDs in the tyres table (1=Supersofts, 2=Softs, 3=Mediums, 4=Hards)
5. Activate the track_model table
6. Go to the name_string_id column and locate the Singapore entry in that column
7. Find the option_tyre column (on the right of name_string_id) and make sure a value of 1 (for supersofts) is on the same line as the Singapore entry. If not, edit it as such.
8. Go to the prime_tyre column (many columns on the right of option_tyre), and enter 3 (for mediums) on the same line as the previous Singapore entry (Hint: Highlight the entire line with mouse).
9. Select any other cell and save the changes (File|Save).
For more realism (visually), this mod changes the hard-coded Softs Yellow Pirelli to the 2013 set of Mediums White (only visually).
(Warning: Due to many different 2013 mods implementations, it could happen to be better for many to edit their own XML text file instead of just copying this mod: See red-color text below.)
Backup your current implementation of the render_materials.xml file you can find in the f1 2012\cars folder so you can restore it after you'll finish a GP or Career weekend in Singapore (for 2013 mods only).
1. Backup your original render_materials.xml file in the cars folder
2. Copy the unzipped cars folder to your f1 2012 installation.
Read the Readme.txt file....
1. Backup your own copy of render_materials.xml in your f1 2012\cars folder
2. Copy the unzipped cars folder found in this mod to the f1 2012 folder.
3. To restore later: copy your own render_materials.xml backup to your f1 2012\cars folder
Members who would prefer to edit their own XML(text) file, read my reply near the end of that Modding Questions page:
2013 Singapore GP Medium White Pirelli Tyres 1.0
For F1 2013 mods installations. Changes Softs Yellow to Mediums White