Here is the same clip, but with a rotational spring. Also a bit more extreme on the sliders.

I also think the most realistic "Hand-held Camera" would be the approach of René aka "the director1974" (ALIEN - the Message) as 4u2ges mentions...
But surely will need a bigger amount of work, will be more elaborate/lavish to do so.
The approach of Rampa is interesting and i will try this out in the next days...


Toystorylab on Vimeo :     Crassitudes (my "Alter Ego") on Youtube: 

This is beautifully simple I gotta try this.

I'm itching to get to iClone but so much to do.
One thing I want to try is what if the antenna itself was the look at and parent to the camera.
I never tried a spring as a twisting effect. I guess it would have to be tiny to limit its effect.
I think this idea might have some weird side effects.
 

A quick Test according to Rampas suggestion...
Used "Antenna01" from Template Folder "Spring", attached to my character, camera look at "Spring04"...
Antenna scaled in Z-axis to 300, spring sliders:





Toystorylab on Vimeo :     Crassitudes (my "Alter Ego") on Youtube: 

@toystorylab: That looks pretty convincing to me. Something to add to the toolbox.

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That was my next goal!

The springs do not twist, so to do this you need a rotational spring turned on its side. Two bones are all you need, with the camera linked to bone 2. Make sure you have the axis you want pointing at the target be negative Y. I prototyped with an antenna spring changed to rotational. If you use the linear spring, each bone rotates to look at the target, so you lose the effect. The rotational behaves in a more standard way where each bone rotates off the rotation of the previous one.

I appreciate the "spring" experiments.  Good stuff.
Looking at TSL's video, I'd like to remove about 90% of the vertical motion and then I think it would be awesome.

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iClone 7... Character Creator... Substance Designer/Painter... Blender... Audacity...
Desktop (homebuilt) - Windows 10, Ryzen 9 3900x CPU, GTX 1080 GPU (8GB), 32GB RAM, Asus X570 Pro motherboard, 2TB SSD, terabytes of disk space, dual  monitors.
Laptop - Windows 10, MSI GS63VR STEALTH-252, 16GB RAM, GTX 1060 (6GB), 256GB SSD and 1TB HDD

Here is a spring delayed tracker for download and use. Think of it as a tripod head with adjustable delay. Now you don't need to attach anything to your character for the effect!
Have a look at the camera movement.


Load it in your scene. It is a blue arrowhead.
Create a camera and link it to the tracker. Then set the camera linkage to be to the last bone end of the tracker in the modify panel. Use the radio buttons to quickly align your camera to the proper location.
Right-click on the tracker and select "look at". Pick your chosen target. The tracker will point at and track the target. The camera will lag a little bit, and then catch up.
You can adjust the amount by playing with spring sliders of the tracker, or by translating the camera forward or back to increase the moment arm.
Have fun!

I knew this would be a fascinating thread and quite amazing results.
I will try one of these camera rigs tonight.