If the part number of the panel is something that has the numbers 30/40 in it, then it is designed to accept ten tandem breakers, usually the bottom five stabs. On some brands, Siemens, Sq D Homeline, Murray, CH/Bryant panels, and a few others, the stabs are notched to accept the tandems. On Sq D QO panels, the positions that accept the tandems have a recess that allows the protrusion on the bottom of the tandem to fit into it. On GE there is a funny U shaped stab welded to the bus bars in the places where the tandems will fit (in addition to the regular stabs).
There are tandems that will fit panels that do not have notched stabs. These are only allowed to be fitted to panels manufactured before a certain date, sometime in the 1980's, before the code changed.
Here is a Square D QO 30 space, 40 circuit panel that accepts ten tandems.
Here is a depiction of the notched vs un-notched stabs and the limiting vs non-limiting tandem breakers.
Here is a diagram on a door that shows where the the tandems are allowed to be installed in the panel.
Here is a Square D QO tandem breaker. Note the hump on the bottom side of the breaker (the breakers are inverted, the bottom is up in this pic). If the panel will NOT accept tandems, this hump will NOT fit and the breaker will NOT snap down, if the panel DOES accepts tandems, there is a recess that lets the hump fit in place and the breaker DOES locks down on the stab.
Charles