I got the little 28 gauge from Joe Schvimmer today and

a double rifle (which most do) although many hate a single trigger gun. Mainly that is because if you have a lion or Cape Buffalo coming at you and for some reason when you pull the trigger nothing happens, with a single trigger gun you are screwed. With a double trigger gun you simply move your hand back and pull the back trigger, hopefully getting your ass out of a bind. Normally recoil of the first barrel cocks the other barrel and it just takes a second to pull the back trigger.
A single trigger gun can be cocked by hitting the butt of the gun lightly against the ground or even hitting the butt as hard as you can with your hand. But in a tight situation with a bear 10 feet from you you simply don't have the time to do that. I have several single trigger guns but I don't have that situation crop up mainly that is because I only use handguns in all my hunts. But in those hunts a TC can cause you to be lunch for a bear or lion as you only have one barrel and no time to open the gun, insert a new round and close it. The length of time it takes to do that can easily get you eaten.
A PH I know that has a 500 Nitro as his back up double gun uses the 2 trigger gun with a standard soft point in the right barrel and a solid in the left barrel. I have taken 3 Cape Buffalo with my 450 GNR in a TC and in each situation the PH told me to use a soft point bullet first and then if I have screwed up to give him the solid.
On my lion hunt my PH preferred a jacketed soft point which was what I used. On something like a lion or other like sized animal I wouldn't think a JHP would be the best way to go. I guess it would be all according to the situation.
There are 2 ways to go when hunting a large dangerous animal or simply a large animal that is hard to put down. Those two choices are a soft point or a solid. In the pic the first two were from my 450 GNR and were a kudu on the left and a Zebra next in line. Neither are obviously not a dangerous animal but my 450 GNR was what I had in my hand at the time. Both were with 500 grain Hornady solids and they took 2 or 3 rounds to put them down. The middle bullet is a solid from another animal and a different brand of solids. The bullet on the right is from my first Cape Buffalo and it was a Hornady 510 grain jacketed soft point in my 450 GNR and it took one shot to put him down. The bullet did exactly what it was supposed to.

Capstick, a self proclaimed expert on African hunting of dangerous game and professional hunter (of which he was neither) often mentions solids that failed by hitting a bone and bending into an L shaped bullet. I have asked several PHs about that and none of them have ever had that happen. I have had friends who have never been to Africa but are "experts" on African hunting that tell all about that happening with large animals. In taking 94 African animals including several large boned animals like Cape Buffalo or Eland I will believe it when I see it.
But back to the single or double trigger guns. My recommendation is go with what you are comfy with. My new double trigger side by side is something I will have to get used to.

. . . that right barrel is "modified" and the left is "full". Not sure if that's a universal standard but it is true of all the guns I've seen/handled. I have not encountered all the various makes/models of course.