Wobbly Rear End & Sticky Gear Change?

Welcome, Kieran.
First things first - get rid of the postcode in your details. Too easy for reprobates to see where you might be.
An oil and filter change may help with the gear changes. I went from Halfords own to Shell and will go back to Halfords next time.
Put the bike on its centre stand, grab the back wheel with both hands opposite each other and see if you can feel any play by pushing and pulling. There should be none - if there is then a bearing is probably shot.
Change the tyres anyway for peace of mind if you are unsure about them. They may be the cause of you wobbles depending on how worn they are and how they have worn.

 

Just remember there are two sets of bearings at the back - the swing arm and the wheel hub. Shaking at 3.15 may show apparent movement in the hub which is in fact the swing arm so check a vertical 12.30 first to see if the hub has any play. None should be evident and would cause an MoT fail anyway.
Check the tyre age marking front and rear. On the sidewall you will find a 4 figure number in an oval; something like 2310. 23 is the week of manufacture and 10 the year. Tyres go hard with age and flat spot if left partially inflated on a sidestand for long periods. It can take a few miles to warm up and lose. I would change anything over 4 years old, irrespective of wear. The rubber denatures.
Check clutch free play too at the handlebar. You should have a 1-2mm gap before cable actuation. Clutch drag can make for iffy selection as can the wrong oil. Do you know what is in it? You might even have light corrosion on the steel plates that needs use to clean off and then an oil change.
I suspect all your reported issues may come down to disuse.

 

Hi, welcome along. New tyres should help improve how the bike feels, just remember to give them a couple of hundred miles to scrub in properly.
Where abouts in South Yorkshire are you, I can recommend a good little independent bike shop to give the bike a check over if you think it needs it, they also do tyres, mainly maxxis, which are superb and great value, but they'll fit any make.

 

Rear suspension linkage?

 

Hi and welcome to the forum

 

Hi Kieran and welcome in to the Triumph Asylum - you'll soon see why we call it that !!

Re your wobbly back-end (????????) have you checked the tyre pressures ?

According to your Owners Manual - 36 psi at the front; 42 psi at the back.

Incorrect tyre pressures on a bike can make an enormous difference to how the bike feels.

Here's a link to a free download for a service manual for your bike : http://www.nirvana-motorcycles.com/Triumph Sprint ST 1050 Manual (2005).pdf