I do a couple different things that are compact and will not draw attention. This is to avoid the potential hassle of being questioned about the use of a professional camera / lens and full size tripod; which may make them think you are trying to reproduce their content or interrupting other users.
So my general workflow / variation:
1) Clean: The first thing I do is bring lens cleaner wipes (or ask for the locals cleaning kit as they may have one they specifically want to be used, if they hand me Windex, Pledge, or 409 I use mine) to ensure the exterior of the lens and the plates the film runs through are clean. I also use bring baby wipes and clean the projection surface to remove artifacts from the background of the projection surface.
2) Stabilize: The main thing I have been challenged by besides warping distortion is shadowing from the light of the projector which on some models of microfilm readers makes using a full size tripod & camera difficult. I have tried the normal style flexible Gorilla Pod attached to the top of the microfilmer well as suction cup style camera mounts for both handheld cameras as well as my smart phone. I have not tried it but there is a also the C-Clamp with attached Gobi-Arm which could be used to manuever around the project head someone might be able to try for better framing using a point and shoot camera.
Though unless I plan on doing a lot of duplicating such copying an entire section of microfilm vs. random snaps for offline review I mostly use a braced handhold shot (I lean my arm or camera up against side or on desk). I like getting a good shot, but also try to avoid getting caught up in the photographers 'everything must be just right' in setting up frames for expediency and minimize distraction to other users.. as in in reality for purpose of capturing the content of many microfilm documents for offline review you likely do not need a 'perfect' copy and there is post process cropping if not perfectly framed or exposed. It needs to primarily just be readable.
3) Camera / Image Format : Since more recent camera phones I have actually had pretty good luck using my iPhone 6plus to capture images quickly and discretely; though I do usually take my point and shoot camera and shoot the images in RAW format to give me the most post-processing flexibility with minimal loss of quality. RAW is not an option on most point and shoots and high quality JPEG is usually sufficient for most when shot on an 8 megapixel or greater camera manufactured in the last 5 years as you are likely never going to print / view the images taken of microfilm bigger than a 8x10 sheet of paper.
4) Frame / Composition: Composing I level the image and frame it using the optical zoom (do not use digital zoom) controls for the field of view of the camera / camera phone and then have my camera up near the projector head. I frame for , if easily done, 1 page per image but still have had good luck post-processing OCR' 2 pages per image.
5) Capture: I go for minimum depth of field of F4 or 5.6 but if possible without a flash F8. To do this you need to put the the camera in Manual mode OR Aperture priority and go for slightly longer exposure time. F2.8 will result in likely not very sharp images around the edges.
If using a camera phone applications commonly go for Shutter Speed priority and will crank down the Aperture for low light conditions so you likely will need to use a non-out of the box camera application (like Camera+) that gives you some manual control over the camera settings.
Results will vary for the Aperture setting by camera and the lens combination but this generally should be a safe setting but expect on most that images may not be as sharp along the edges. Being this is a projected image you obviously need to ensure your camera flash is always disabled.
5) Post Process: After taking the image I process the image through one of the camera apps (i.e. Prizmo does ok OCR) or preferably wait until I get home and then run it through Adobe Lightroom to adjust exposure and contrast, and usually crank up the blacks a bit if it is text. I only sometimes feel the need to convert it to black and white or attempt to correct any warping if present (as OCR still seems to work). I then go into the print module of Lightroom and print it to PDF and then use the recognize text function of Adobe Acrobat to OCR the document to have a text searchable PDF version of the document.
a) For Image duplication I have to vary the technique because I want as good of copy of the image for actual retention and placement in a tree vs. text is likely just for review and reference. These I will alway if I have it almost always use some sort of camera stabilization device (even if it is just holding my phone up against the frame of microfilm reader) and then later having to crop the images if not fully zoomed in (I never use digital zoom) in a camera application or Adobe Lightroom.
b) If the Microfilm reader is fitted with a printer (and it is is decent working condition without a lot of artifacts) and the document is important I will actually generally use that for text vs. trying to take my own image and then scan it when I get home at least for text as I can scan and OCR in a single step.
Note: One thing to keep in mind is that the LDS Family History Library, at least my local one, and my local Public Library have signs up about no duplicating the content of their microfilms. I have only seen a couple people called out on but you do not want to be that person that gets banned so not making a big production of your photographing is probably important as they do not want everyone doing it; it it also ties up their readers. Though personal use photography can probably be considered 'fair use' to capture images since they have printers/copiers connected to some of their microfilm readers.