I've had good luck #1 and #2.  I switched to using the stay fresh produce bags.   For me, the bags seem to make the difference.  The ones I use were from the dollar store.  There were about 8 bags in the package (the green  twist tie version works better). 

My primary method this year has been #3. I used baby bottle liners instead of plastic bags-they were readily available. However, with the total number of cuttings I worked with, I have also used cups. With one particular set of cuttings I reverted to just burying them in a pot of potting mix and letting them ride the winter outside.

I do #2. i did #1 first and was as successfull but still got mold occasionally even though washed cuttings with cleaning solution, annoying to wrap and unwrap 60 cuttings checking for mold/replacing newspaper.
Its nice to be lazy and just throw the cuttings in the damp moss and forget about it until it roots. Only challenge i remember is just have to be careful removing rooted scions from the clingy moss, (hard also removing as there might be other scions with some roots in the same bag that you dont' want to remove yet).
Once rooted, I put in a plastic cup with drainage holes burned in bottom and sides with a soldering iron (I also first cut the cups in half so can easily transfer to gallon container later without breaking roots and rubber band the tops to keep the cups from breaking like so). I used to put that cup within another cup to avoid rubber band, but the holes wouldnt line up and get clogged/waterlogged too easily (not sure but you can see the cups within cups here). I used red cups around the plastic cups to avoid sun/light damaging the roots. It worked out well last year, had too many/90% cuttings root hehe.

" With cups...for me...they are constantly falling over and spilling their contents.

Low tech and cheap. Two things I like a lot."

Don't you just hate when that happens! This was my low tech/cheap solution to the problem....

http://figs4funforum.websitetoolbox.com/post/Quick-amp-Easy-9Pack-Tutorial-5684260

I bury my cups about 3/4 of the way in my raised bed garden.  I have only lost one this year, and it was a wee tiny twig.  

I swear on them '20J16' foam cups, and nobody will convince me otherwise...

Are those cups strong enough to use for more than one season Gorgi? Do they tend to deteriorate?

When my cuttings show enough root in the sphagnum, they go into two liter plastic soft drink bottles, with soil mix, holes in the bottom sides for drainage/air flow and the top cut so it acts as a humidity dome.  Seems to work and is a no cost option since I get all my bottles from friends or the garbage.

Grant
z5b
 

This is why I moss...with greater than 95% confidence usually within four weeks. The work is the next 12-24 months nurturing that pencil or sharpie with roots to an established vigorous tree that will yield fruit.

I'm new to fig rooting, but for my cuttings went with #4. I've rooted hundreds of roses, blueberries, etc and have many black plastic bands, so I used those. (see photo below - not mine, and obviously not a fig. :) ) 

I used a mix of half perlite, half planting mix.  And the bands with cuttings were placed in bins covered with plastic (opened daily). Since they are square, they pack nicely and are stable. The bins are in a warm room with a temperature range of the mid 60*F at night, up to about 80*F in the daytime on a warm day. No sun hits the boxes.

Assuming I get rooting, the downside is I won't be able to see what's happening unless I tip them out or am able see roots through the mostly open bottoms. But I also won't have to transplant anything until they are actively growing and ready to go into larger pots.

When the cuttings begin to leaf out and need light, I'll move them into covered bins outside in the shade. Warm Socal area here.