I usually advise companies to authorize around 10 to 15 million shares of common stock. Around 8 or 9 million shares are issued to founders with a 1 million to 2 million share option pool, for a fully-diluted base of around 10 million shares. The remaining authorized but unissued shares are a reserve in the event more shares need to be issued.
From a purely mathematical perspective, it doesn’t matter whether there are 1 million or 10 million fully-diluted shares. However, when companies are granting options to new employees, even the smartest engineers feel better receiving options to purchase 100,000 shares as opposed to 10,000 shares, even if it represents the same percentage ownership of the company.
Assuming a $15/share IPO price and dilution due to financings, 20 million shares outstanding will result in a $300M market cap, which is about the minimum size necessary to complete a successful IPO. This avoids having to do a reverse or forward stock split at the time of an IPO.