First of all, diamonds are not as rare as people think. The perception that they are rare is perpetuated by heavy advertising and marketing. Billions of dollars are spent to keep demand high. It is no secret that the prices of diamonds are also kept artificially high through tight supply chain management, ie. a limited number is released to a select few distributors who in turn re-sell it down the supply chain. The exceptions? Fancy colored diamonds, large sized diamonds with good color, clarity and cut grades, phenomenal diamonds (color-change) etc.
Back to your question. It's hard to give a straight answer because it really depends. There are so many variables! Size, Quality, Origin etc...
To make things simple, I'll just put it this way - an extra, extra fine sapphire (ie. top quality) from Kashmir or Burma is rarer than the majority of D Flawless diamonds; and a mediocre sapphire is as rare as a mediocre diamond.
Gemologists do not differentiate gemstones along the "precious" "semi-precious" line because it is grossly inaccurate. For instance, tourmalines are usually considered "semi-precious". However, a fine Paraiba tourmaline is more expensive than many rubies and sapphires. In this day and age, the "precious" "semi-precious" classification is just meaningless. ALL gemstones are precious. In particular, I think many fine colored stones such as Tanzanite, Tsavorite, Jadeite, Red Spinel and so on, are much rarer than diamonds in general.
I LOVE gemstones and I'm a gemologist