OK, let's go through a process of selecting a transistor.
(I note that we're not using a constant current driver despite my strong recommendation.)
1) Firstly go to
digikey. (Yeah, even if you don't plan to buy from them.)
2) in the search box type "mosfet".
You get a list of broad categroies that might contain what you're searching for. Generally the one with the most hits is the one you want.
3) click on "FETs - single"
Now you're into the parametric search. We first need to start narrowing down by some obvious first choices:
4) Select from the "FET Type" box, "N Channel, metal oxide" and then click Apply filters.
5) Scroll to the right, and withing "Mounting type" select "Through hole" and click apply filters.
Your power supply is a low voltage, so you'll need mosfets which can be turned on with a low gate voltage.
6) In "FET feature", select "Logic Level gate" and click on apply filters.
There are still over 1500 devices, so lets narrow down on some ratings that may be useful.
7) in the "Drain to Source voltage (Vdss)" box, select all values between 20V and 60V. Then click Apply Filters. Whilst a Vdss outside this range would be fine, it's useful to have some margin at the low end and at the upper end the devices will be more expensive (all other things being equal).
The next important thing is the current. Although we want 350mA, I wouldn't pick anything capable of less than 1A. At the upper end, let's say 15A?
8) in the "Current - Continuous Drain (Id) @ 25° C" column, select all values from 1A to 15A and click Apply filters.
We don't want something fragile, so let's pick a device that can handle at least 2W of dissipation. and maybe up to 50W.
9) in the "Power - Max" column select all values between 2W and 50W and click on apply filters
OK, now I show a little over 40 devices, and that's small enough -- any fewer and I might open up my selections a little (and probably at the upper ends).
10) In the quantity box below the search boxes, enter "1" (or a greater number as appropriate) and submit it.
11) in the results table at the bottom, look in the price column and click on the small blue triangle at the left of the column to sort in order of lowest to highest price.
The first result is
this. It's cheap and would do the job.
However, if you wanted to bolt the device to a heatsink, a few rows down is a TO-220 package
here.
If you didn't want to have to worry about insulating the device from the heatsink, there's a full plastic version a row or so below.
Here.
If you're not intending to purchase from, digikey, then perhaps you would take the part numbers from the first page of results and search for them on your preferred supplier's page.
In this case, we chose the specifications with plenty of excess capability, and we know that there are unlikely to be thermal issues, so we can probably go ahead.
In other applications (especially high current applications) there may be a number of other considerations.