The most common problem with it starting in singleplayer is due to the installer for 0.3z R2. By default the installer points to the location "C:\Grand Theft Auto Vice City" which is usually not where your Vice City will be installed. Players are either unaware that it has to be installed to this folder or they don't know where there Vice City folder is. Therefore they click next and the installer just extracts the files into an empty folder.
The player will then probably open up the server browser assuming everything is ok and when they try to launch for the first time a message will appear saying "Please locate your gta-vc.exe" they will then locate it and the game will launch. The problem is however is that since the extracted files are not in the Vice City folder (mainly vcmp.flt in Grand theft auto vice city/mss) the game just loads up normally to the singleplayer menu and the person is confused.
Alternatively I think having a .dll in the root folder of Vice City e.g a mod that uses a .dll file as a component would be the speedometer/tachometer for Vice City mod. That can also cause the game to launch into singleplayer if I recall.
If you are someone reading this and you have the same problem and do not know where your Vice City folder is then it will either be...
CD/DVD installation: C:\Program Files\Rockstar Games\Grand Theft Auto Vice City
OR
Steam games installation: C:\Program Files\Steam\steamapps\common\grand theft auto vice cityGood luck.