Generally, we have: SR transceivers with multi-mode fiber and LR with single-mode fiber. Is it possible to use single mode fiber with 10G-SFP-SR transceivers ? And same question for LR: is it possible to use multimode fiber with 10G-SFP-LR transceivers ?
No i would not recommend using multimode with singlemode cabling and vice versa. MM to singlemode cabling is probably the worst and might not even work, but SM to multimode cabling might give you a short range because it lets out a lot of light at the edges.
Instead i would use 10GBASE-LRM, (long reach multi-mode) originally specified in IEEE 802.3aq is a port type for multi-mode fiber and uses 1310 nm lasers. It allows distances up to 220 metres (720 ft) on FDDI-grade multi-mode fiber and the same 220m maximum reach on OM1, OM2 and OM3 fiber types.
Most 10GBASE-LRM transceivers also allow distances up to 300 metres (980 ft) on standard single-mode fiber (SMF, G.652), however this is not part of the IEEE or MSA specification.
You should always use the correct fiber type for the connection required. If you need to run new fiber you need to think ahead of what might be coming. Generally, it doesn't pay to save on fiber as a redeployment a few years later is expensive.
If you have to put up with old fiber and need to squeeze out more performance please add the appropriate details to your question.
No, you cannot use SR optics with single-mode fiber. You'd be firing a 50/62.5 micron beam at a 9 micron hole; at best 18% of the light will enter the fiber.
Yes, you can use LR optics with short, high quality multi-mode fiber with the use of a "mode conditioning cable". That special section of cable is the Magic(tm) that makes it possible by precisely aligning the 9 micron laser with the 50/62.5 micron fiber. [This procedure is not recommended.]
I think using a MM fiber with a LR transceiver or a SM fiber with a SR transceiver would adversely affect performance, regardless of the form factor (SFP, QSFP, CFP, etc). The technician/engineer running the fiber for you should be able to confirm that. Normally, I would use a LC duplex or LC-SC fiber for LR-type optic and MPO/MTP fiber connectors for SR-type optic. The number of fiber strands you need for the SR-type optic would depend on the number of optical lanes configured on your transceiver. For SR4 (4x10G or 4x25G), you would need a 12-strand MPO/MTP fiber whereas for a SR10 (10x4G or 10x10G) optic, you would need a 24-strand MPO/MTP fiber. The most widely used operating frequencies are 2.0MHz (OM3) and 4.7MHz (OM4). Another spec to look out for is key-up/key-down or key-up/key-up.