Suppose you had a single sheet of graphite the size of an A4 page. What would it feel like to handle it?
Would your fingers instantly tear through it as you tried to grasp it?
Could you punch your finger through on purpose?
Could you tear it like a sheet of paper?
Could you fold it into a paper aeroplane without breaking it?
I will assume that by "a single sheet of graphite" you mean graphene.
The link in the comments to the wikipedia page on orders of magnitude of different forces, provided by @Alchimista in the comments, is helpful in getting a perspective on at least a few of your questions. Two useful reference values are the force exerted by a 1-kg weight due to gravity at sea level, ~10 N, and the force required to rupture a typical covalent bond, ~1 nN.
Perhaps the first question would seem to be the easiest to answer, but I found it is not: mechanoreception by sensors on skin is complicated. In the same way you can sense very gentle contact, you may be able to sense the graphite sheet touching your skin. It would probably be an odd feeling. Note also that the sheet would be just barely visible.
The strength of a material is described by more than one property. Perhaps the simplest property to describe is the tensile strength, the stress required to rupture the material by pulling it apart under tension. You can use estimates of the geometry of a sheet of graphite and the C-C bond enthalpy to arrive at the force per unit length of the material require to rupture it:
$$\pu{F_l(N/m)} \approx \frac{\pu{350 kJ/mol}}{(\pu{2.46 Å*1.42 Å})}$$
That's $F_l\approx\pu{16 N/m}$, so you might expect a single layer of graphite that is 1 meter long to be able to hold up a 1-kg weight suspended normal to this dimension, assuming the force is evenly distributed. If we assume the sheet is ~$\pu{2 Å}$ thick then we arrive at an estimate of the tensile strength of ~$\pu{10^11 Pa}$, making it the "strongest material ever tested."
Here I've estimated the number of bonds per unit length normal to the direction of tension as 1 per 2.46 Å based on this figure depicting the dimensions of a sheet of graphite. I also assumed the average force required to break one CC bond is $\Delta_{bond}H/r_{CC}$ and used a low estimate of $\Delta_{bond}H$. You could derive a similar value of $\Delta_{bond}H$ from the heat of sublimation of graphite.
Based on this information I would assume that it would be easy to tear the sheet, but that you would not necessarily tear the sheet by grasping it, provided you did this very carefully. On the other hand, you could certainly punch a hole with your finger. Ignoring subtle issues regarding the geometry at which forces are applied, we can use the previous estimate of the rupture force per unit length and an estimate of the radius of a finger (~1/2 cm) to arrive at the force required to punch the sheet, or $\approx\pu{0.5 N}$, or equivalently the pressure:
$$p=\frac{2F_l}{r_{finger}}$$
or $\approx\pu{7 kPa}$. A fat cat would only exert a pressure of about 300 Pa.
Could you fold it into a paper aeroplane? Apparently graphene is tough enough to be significantly bent out of shape without rupturing (Refs. 1 and 2), so I'd have to guess "yes".
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