I was thinking that Normal is the most overlooked type of all, besides Bug. Here are some Normal Pokemon I decided to put into my team.
At a Glance:
In Depth:
Swellow (M)@Toxic Orb
Ability: Guts
Nature: Jolly
EVs: 252 Attack, 6 Sp. Defense, 252 Speed
-Brave Bird
-Facade
-U-Turn
-Protect
This is pretty straightforward. Use Protect to activate Toxic Orb and Guts, then use Facade to attempt a sweep. If I think I won't be able to do much, U-Turn to get out while doing some damage. Brave Bird is to hit Ghosts and others Facade won't hit very hard. With both Guts and Poisoning, Facade reaches a power of 210 (I think) and maxed Attack and Speed tries to hit hard.
Togekiss (F)@Lum Berry
Ability: Serene Grace
Nature: Modest
Evs: 252 HP, 152 Sp. Attack, 104 Speed
-Aura Sphere
-Air Slash
-Nasty Plot
-Roost
This is a somewhat bulky special sweeper. It can take a hit or two to set up a couple Nasty Plots, then sweep with Aura Sphere or Air Slash. Serene Grace increases Air Slash's flinch rate, which is good, and Aura Sphere hits what Air Slash doesn't, like Rock or Steel. If health gets low, I use Roost to restore health.
Miltank (F)@Leftovers
Ability: Scrappy
Nature:Impish
EVs: 252 HP, 136 Defense, 120 Sp. Defense
-Curse
-Body Slam
-Heal Bell
-Milk Drink
I love this. It requires some setting up to do (using a bunch of Curses), but if I pull it off, the results are huge. After Curses, I use Body Slam. If my HP gets low, I use Milk Drink. If statuses are plaguing my team, I use Heal Bell. Simple as that. Scrappy hits Ghosts as well. It can take many hits, especially coupled with Milk Drink. After Curses, Body Slam does a lot, even to Rock and Steel types. If a strong special attacker comes in, I just switch to Snorlax. The 30% paralysis chance from Body Slam helps, and so does a little surprise factor.
Lickilicky (M)@Life Orb
Ability:Own Tempo
Nature: Brave
EVs: 116 HP, 252 Attack, 140 Sp. Attack
-Explosion
-Fire Blast
-Ice Beam
-Earthquake
Many don't expect this. The main purpose of this is to break down walls and for attacking diversity. Lickilicky can take a few hits with 116 HP, and kill with Earthquake, Ice Beam, and Fire Blast (*coughSkarmorycough*). If Lickilicky can't go on anymore, Explosion. It seems pretty unusual, but works fairly well.
Staraptor (M)@Sharp Beak
Ability: Intimidate
Nature:Jolly
EVs: 4 HP, 252 Attack, 252 Speed
-Brave Bird
-Close Combat
-Roost
-Substitute
This aims to attack but with a bit more durability. Substitute it to avoid status, while Brave Bird and Close Combat hit hard. Roost is to recover damage from Brave Bird and from defense drops. Simple enough. (Note: I just found this to fill in the sixth spot, so if anybody has something better other than Blissey or maybe Porygon-Z, I'll consider it.)
Snorlax (M)@Leftovers
Ability: Immunity
Nature: Adamant
EVs: 4 HP, 252 Attack, 252 Sp. Defense
-Body Slam
-Earthquake
-Fire Punch
-Crunch
This is an offensive Snorlax that functions well enough as a Special wall. Maxed Attack and Sp. Defense ensures that the attacks hit hard and that Snorlax can take the hits. Crunch hits Ghosts that think they can switch in. Body Slam for STAB and paralysis. Earthquake and Fire Punch is for type variation. At first the ability was Thick Fat, but after some Toxic Spikes problems, I decided to change it to Immunity.
Conclusion:
So, there it is. Again, I wanted to use what I thought was a very overlooked type. Maybe I just want to make a point. Oh well, it doesn't matter. On Shoddy Battle, this team has had above-average success, mostly wins but more than a few losses. I have had trouble with Stealth Rock, Spikes, and Toxic Spikes, but I don't know of any Normal type with Rapid Spin. I also don't have any that set them up either. I have also had trouble with weather teams, like a rain team with Kingdra killing my entire team with just two Pokemon. Of course, there is a Fighting, Ice, Electric, and Rock weakness, but somehow, I haven't had any major problems with that. If Fighting comes in, I switch to one of my flyings, and if Ice, Rock, or Electric come in, I switch to my relatively bulky pure-Normals. That's about it, rate please!
At a Glance:






In Depth:

Swellow (M)@Toxic Orb
Ability: Guts
Nature: Jolly
EVs: 252 Attack, 6 Sp. Defense, 252 Speed
-Brave Bird
-Facade
-U-Turn
-Protect
This is pretty straightforward. Use Protect to activate Toxic Orb and Guts, then use Facade to attempt a sweep. If I think I won't be able to do much, U-Turn to get out while doing some damage. Brave Bird is to hit Ghosts and others Facade won't hit very hard. With both Guts and Poisoning, Facade reaches a power of 210 (I think) and maxed Attack and Speed tries to hit hard.

Togekiss (F)@Lum Berry
Ability: Serene Grace
Nature: Modest
Evs: 252 HP, 152 Sp. Attack, 104 Speed
-Aura Sphere
-Air Slash
-Nasty Plot
-Roost
This is a somewhat bulky special sweeper. It can take a hit or two to set up a couple Nasty Plots, then sweep with Aura Sphere or Air Slash. Serene Grace increases Air Slash's flinch rate, which is good, and Aura Sphere hits what Air Slash doesn't, like Rock or Steel. If health gets low, I use Roost to restore health.

Miltank (F)@Leftovers
Ability: Scrappy
Nature:Impish
EVs: 252 HP, 136 Defense, 120 Sp. Defense
-Curse
-Body Slam
-Heal Bell
-Milk Drink
I love this. It requires some setting up to do (using a bunch of Curses), but if I pull it off, the results are huge. After Curses, I use Body Slam. If my HP gets low, I use Milk Drink. If statuses are plaguing my team, I use Heal Bell. Simple as that. Scrappy hits Ghosts as well. It can take many hits, especially coupled with Milk Drink. After Curses, Body Slam does a lot, even to Rock and Steel types. If a strong special attacker comes in, I just switch to Snorlax. The 30% paralysis chance from Body Slam helps, and so does a little surprise factor.

Lickilicky (M)@Life Orb
Ability:Own Tempo
Nature: Brave
EVs: 116 HP, 252 Attack, 140 Sp. Attack
-Explosion
-Fire Blast
-Ice Beam
-Earthquake
Many don't expect this. The main purpose of this is to break down walls and for attacking diversity. Lickilicky can take a few hits with 116 HP, and kill with Earthquake, Ice Beam, and Fire Blast (*coughSkarmorycough*). If Lickilicky can't go on anymore, Explosion. It seems pretty unusual, but works fairly well.

Staraptor (M)@Sharp Beak
Ability: Intimidate
Nature:Jolly
EVs: 4 HP, 252 Attack, 252 Speed
-Brave Bird
-Close Combat
-Roost
-Substitute
This aims to attack but with a bit more durability. Substitute it to avoid status, while Brave Bird and Close Combat hit hard. Roost is to recover damage from Brave Bird and from defense drops. Simple enough. (Note: I just found this to fill in the sixth spot, so if anybody has something better other than Blissey or maybe Porygon-Z, I'll consider it.)

Snorlax (M)@Leftovers
Ability: Immunity
Nature: Adamant
EVs: 4 HP, 252 Attack, 252 Sp. Defense
-Body Slam
-Earthquake
-Fire Punch
-Crunch
This is an offensive Snorlax that functions well enough as a Special wall. Maxed Attack and Sp. Defense ensures that the attacks hit hard and that Snorlax can take the hits. Crunch hits Ghosts that think they can switch in. Body Slam for STAB and paralysis. Earthquake and Fire Punch is for type variation. At first the ability was Thick Fat, but after some Toxic Spikes problems, I decided to change it to Immunity.
Conclusion:
So, there it is. Again, I wanted to use what I thought was a very overlooked type. Maybe I just want to make a point. Oh well, it doesn't matter. On Shoddy Battle, this team has had above-average success, mostly wins but more than a few losses. I have had trouble with Stealth Rock, Spikes, and Toxic Spikes, but I don't know of any Normal type with Rapid Spin. I also don't have any that set them up either. I have also had trouble with weather teams, like a rain team with Kingdra killing my entire team with just two Pokemon. Of course, there is a Fighting, Ice, Electric, and Rock weakness, but somehow, I haven't had any major problems with that. If Fighting comes in, I switch to one of my flyings, and if Ice, Rock, or Electric come in, I switch to my relatively bulky pure-Normals. That's about it, rate please!