“It would be good to talk about the difference between shares and founders shares. It does matter in the stage where you’re evaluating a job offer, no?”
First a caveat: The following is true (mostly) only for US-based companies. Other countries have very different ways to handle equity (e.g. in Germany it is common to set up your company as a GmbH which can have multiple equity holders but doesn’t issue stock or shares to do so).
Now to the question: No.
The simple matter of the fact is — unless you’re an investor you will always get common shares. The way it works is: A company has common shares, these are the shares a company starts out with and doesn’t come with any special rights (such as liquidation preferences). When a company raises money it will issue new shares which usually are “preferred shares” — preferred in this case means that they come with a set of additional rights such as liquidation preferences and rights for board seats.
When you join a startup you will either get common shares (exactly the same as the founders) or you will get options (with options being just that: The option to buy shares at a pre-determined price). You will not get preferred shares nor are there (usually) any other classes of shares which are special to the founders.
Hope this makes sense — and good luck with that job hunt! :)