I fish both copolymer and braid.
The best and thinnest copolymer is Kamikaze from Oz - the problem is, the shipping costs the same as a 150-m spool.
It's also the only salt-targeted UL copolymer.
Yo-Zuri is the old reliable original copolymer, though also the thickest. I last bought a bulk spool from Amazon.
I've broken off at least a couple massive fish on copolymer, including a lifetime snook - not complaining, it was great fishing. Broke off a dinner-size spec one day just because I hadn't fished 3-lb in awhile, and forgot to apply the light-line subtleties.
Copolymer brands I've tried and don't like include Kastking and Berkley.
Aside from knot strength, I evaluate copolymers on their stretch mode when you nip them or break them off. If it whiskers when you try to nip it, it's too soft (see Berkley) - if it feels rough, it's too hard (see Kastking). What I don't accept about the Berkley is how it bulk-stretched and slid under spool wraps one day when my daughter was catching big sheepshead (on 10-lb ML).
Seaguar is my only choice in fluoro, nylon mono, too - they simply have the very best knot strength (= toughness).
Both are getting harder to find in reel-fill spool sizes, but I buy bulk spools of my favorite compositions - always have Red and Abrazx in my inshore sizes - and Blue in tippet spools covering a wide range (4-30-lb).
Seaguar doesn't call their copolymers by this moniker, but their spendy Tatsu and closer-to-earth Senshi are thin and tough, and I have the latter in a bulk spool.
While it's always a good idea to use your bail manually and turn with your rod sideways to take up all slack before you begin retrieve, it's imperative with braid.
Braid - the very best-behaved braid ever is Florida Fishing Products, which doesn't come in smaller than 10-lb, but believe me, that's small enough.
I've broken both 4-strand Yo-Zuri and 8-strand Sufix, and it made me choose to always look for 8-strand.
Braid I didn't like - again, Kastking - not coated, not tough, and abrasive.
Suxfix 832 is offered in 6-lb, which is the same diameter as 1-lb fluoro tippet.
I don't think any US supplier offers smaller braid, but they do in Japan.
I never tie braid directly to terminal tackle, but always use an improved Allbright knot to fluoro or copolymer shock tippet - something needs to stretch before it breaks.
I also use the lightest-touch of Zap-CA+pink on my splicing braid knots, chased with an accelerator (Mitre Apel spray is convenient).
This process is carried out in a ritual with a loop in the shock tippet hanging through the first guide and under tension from the weight of my spring scale,
and finished by setting my drag to 1/4 of the weakest link - braid or shock-tippet test, or rod max line rating.
This isn't set through the bent rod, but a straight shot to the spool - the dynamics of leverage through the rod combined with jerk can get that 4x back and more, and is how rods get broken and line snapped, even when drag is properly set.
I tie my shock tippet surgeon's loop large enough to loop-on either a weightless cigar cork or titanium bite trace.
For toothy fish, absolutely cannot beat Mako titanium leaders from Russia, shown next to a "normal" size salt swivel snap hook (60-lb), to give you perspective on their diminutive hardware. They're light enough to use on a fly rod for toothy fish.
The advantage of titanium is that it stretches before it breaks (and single-strand titanium will also knot).