How can I verify where the pipes in my basement go?
this is fun, this is straight up detective work, and i'm sure you can figure it out with some guidelines

mike gives good advice, with the sniff test and such, but i think it might be easier another way.

if it was laid out for a basement bathroom, which would be my guess, then those smaller pipes would be vanity drain, and would be laid out to line up within a 2x4 wall.  you will need a framing square, and a 4' level or some equivalent thing to make a perpendicular (square) line off of the wall nearest one of the smaller pipes, and then the other.  basically you will want to draw a line that goes where the edge of a 2x4 would be if that pipe was right in the middle of it.

my guess is, from looking at your pic, that if you did this with both of those smaller pipes, they may not both be on the same line with each other, but the center of that 3" pipe will fall near perfectly 12 1/2 inches from the line you draw, on one of them.

12" is the standard layout for toilet from the wall, so that is your guide.

this may not help, but could.

other things to consider...

these builders are so cookie cutter by default, that asking a neighbor to describe their basement bathroom layout, could result in some answers.

your vanity drain will be a 1 1/2 inch pipe.
your tub drain will be a 1 1/2 inch pipe
your stand up shower must be a 2 inch pipe
toilet can be 3 or 4, but will be more than likely 3, and will then be right about 3 1/2 inches around if measured at the top.

they could easily put a glue type p-trap in the 2 inch or 1 1/2 inch pipe for your shower or tub, and bury it in the concrete, as you only need access if using the non-glue, slip joint type trap.

if i was betting, i would say the pipes you say are 2" are really 1 1/2" inner diameter, and therefor 2" outer, and if you do the square line thing i describe, you will find that from the one line, the other 2" pipe is around 6" forward, and the larger pipe is centered about 12" forward, this would mean your room is roughed in for a vanity and tub/shower. (really odd to have toilet in between, me thinks)

anyway, those lines coming from above are just left long so you can find em.  they are vents for something, i'm guessing plumbing (an extra mechanical room provision would be very odd)  they are vents, and you have much more freedom with vent pipes then with drain pipes, (for instance you can do straight 90 degree turns with no radius with a vent but not with a drain, as the water must be able to sweep to some degree for ease of flow around turns and changes in direction)  so since they cant orient the upstairs plumbing / framing to match the basement, they would just leave you vents to tie into later, and expect you to have the floor joist region to be able to easily make the appropriate turns in the upstairs pipes to match with what you have in the basement.  (remember these builders are doing the minimum required so that you can put a bathroom in later without having to conduct major surgery on your basement floor and / or upstairs plumbing and framing to get a bathroom accomplished at a later date.

move the insulation and see where those pipes from above connect to.  as reasonably certain as i am to what they are, 3 - 1 1/2 vents for a toilet vanity and tub is vent overkill as i see it. (in fact, i'm a stickler on venting myself and to do this right there would have to be a vent coming out of the ground, already tied into the toilet line, so that may be another option, (that one of the 1 1/2 inch pipes are a vent) - (your vent should always tie in below the level of the p-trap, which since your p-trap is built into the toilet, the vent for the toilet should always tie in below that level, and in this case, would be connected to the toilet drain, below the floor, and therefor be coming up somewhere nearby.  (i don't have my book with me right now, but I wouldn't vent a toilet with a shared vent from a vanity)

so yeah, if after drawing that perpendicular line from the nearby walls - you find that these 2" pipes are in line with each other, then my guess is one is a vent for the toilet. and i would be quite pissed off that nothing was written, etc or labeled so you knew which - as more than likely you can't legally just use one or the other.  (one is designed to be a vent and will be tied in as such and the other will be tied in to the 3" as a drain and most likely will be different in build and function.

this exercise is all about you finding out as much as you can and just applying that to some common sense building practices.

good luck, brother.