Hi,
At last you`ve got enough time to apply for visa. If you are going to stay in your friend`s home you will not free them from hassle and paper formalities (believe me). At least they have to register your stay in the local authorities - otherwise you are illegal. Right now in Moscow very harsh new immigration regulations are in power. So ALL foreigners must be registrated unless they stay in hotels and hotel stuff provide this automatically.
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Second point - visa applicatiom. All the details you may obtain in Russian Embassy, Singapore (consular section). As far as I know the necessary things are-
- return air tickets
-hotel booking with confirmation letter. In your case original (not photocopy, e-mail etc) letter of invitation from your friends which they have to compose accordingly.
- properly filled in application forms which you may get at the Embassy
- visa fee payment (how much I have know idea, it depends on your nationality, for indonesians for example its 75USD for single tourist visa).
-I heard nothing about on-line visa application especially with Russian Embasy. Please be careful of frauds.