I love playing Singed but I wouldn't say I'm a main or someone who knows everything about him. I play him for fun and when I just want to have a good time, so this probably isn't the best build, but I like it
I usually start with 
and 
. I start because of it's potion passive (yes not the glory, the potion extension, I'm weird) because I think that it basically is what Singed's passive should be instead of 1/4 mana=hp and start the planning to turn it to 
because of sustain and synergy with .
Probably the most important thing about Singed is DO NOT BUILD A PURE MANA BUILD JUST BECAUSE OF HIS PASSIVE. As Admiral Ackbar would say, It's a Trap! While it's usually fine to build some item like with mana as part of it. It shouldn't be your main goal.
You are correct on rushing 
then getting 
. the synergy of these items with your poison will make anyone realize why they shouldn't break rule #1.
With boots, I'd always get 
unless they have someone like riven or yi in which case you take 
. The reason for getting is the tenacity it gives you.
My main MR item is 
due to the increase to healing. My main Armor item is 
due to the speed increase although you can also build 
for anti-crit and anti-attack speed.
If you plan on split pushing then you might want to get 
before and get a 
eventually. You still want the to help you escape.
I take 18 in Ferocity and put the rest in whichever (not lvl 30 yet so I can't do 18-0-12 or 18-12-0). Take Sorcery for damage, I take double edge sword because of the 3% increase on poison (I'm all about full damage), take natural talents because lifesteal is worthless on your poison and you won't be doing enough damage with everything else. I take oppression due to it being safer than bounty hunter and once you get you will always be using this. Take piercing thoughts for magic pen. Then lastly take DeathFire's for the extra bit of damage to poison and fling.
Putting the other twelve in Resolve will probably be the safer of the two, put the first 5 in unyielding due to it usually coming out better than recovery. However, I don't know if you took all health regen as well as runic armor and if recovery would come out as better. I'd take Tough Skin next, unless you are planning on running away through the river, than take explorer. Take runic armor next for the bonus regen on your ult and in general. Then take insight unless, again, you are fully built on health regen (again, I don't know how well that will turn out).
Putting the 12 in Cunning is for more damage (so it's what I am going to do), put the first 5 in savagery for when you have to hit a minion with a basic attack. I don't think wanderer is good because so long as your poison is on someone, it won't be helping you. Take either Secret Stash if you want even longer potions (with you will really be getting your money out of those potions) or take assassin because you are in a solo lane for more damage (plus if you are running away or split pushing, you are probably not with your team). Take either merciless if you want, you guessed it, more damage, or take meditation to keep yourself in out of base for longer. Then take dangerous game because you won't get a lot out of Bandit, granted you can take it if you want more gold gen at the cost of some health and mana back.
Sorry if this went on for too long, Thank you for reading and I hope this helped you, hope you have fun helping more people break the number one rule!
then I usually buy 
and start my 
or 
but I'm honestly not sure how effective it is on singed, the mana converts to some hp, you get a lot of armor which is nice...but CDR isn't typically something I've seen built on singed, despite how important his ultimate is to his teamfighting.