Hello Rav,
My son was very disturbed when he found a spider right next to his bed on Shabbat, which prompts the following question:
If a non-dangerous spider or ant is in your house on Shabbat and it is upsetting you (or the children), is it permitted to trap it briefly in a cup with a tissue over the top of the cup and take it to be released outside? That would be the best option, but If not, could you try to get the bug onto a piece of paper or the like and then quickly take it outside. (both are indirectly handling the muktzeh, but in the latter option the bug is never really trapped).
Additionally, is there any difference if it is a mosquito or bee?…both potentially could bite and be very annoying (assuming there is no one actually allergic to the bite, so the bite is not life threatening, just very annoying). Any difference for sephardim?
Thank you.

A. If the spider is non-dangerous one should not trap it or kill it. The best and easiest way to deal with it is to have an open door or widow ready, then have it climb a large piece of paper or cardboard and promptly throw it out. Since the spider can easily drop from the surface, it is not trapped. There is also no prohibition of muktza involved since it is “tiltul min hatzad” or indirect and unusual moving. Some Poskim consider insects also as a “geref shel re’i” or a disgusting refuse item. (See Pri Megodim 316; 8, Kaf Hachaim ibid. 74).
As far as bees and mosquitoes are concerned, it is best to spray on one self repellent, or without trapping or killing them, attract them to a plate or surface with honey or similar and proceed as above.
Horav Shlomo miller’s Shlit’a opinion is similar.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a