I've been playing Lucas competitively since the game launched. I am very familiar with some matchups. Each thing I've commented on, I've played against another competent player several times.
Lucas Vs
Falco, advantage: Falco. Falco is fast enough to close distance very quickly. Constant laser shots refresh Falco's A and are very hard to absorb. Shine deflects spells and deals damage.
Ice Climbers, advantage: neutral. Lucas has better projectiles and will force IC to close distance. Lucas's u-smash will devastate either Nana or Popo. ICs can combo Lucas effectively. Ice projectiles cannot be absorbed.
Ivysaur, advantage: Lucas. PK Fire equals razor leaf in terms of priority, but PK Fire casts much faster. Ivysaur will have to resort to other attacks. Lucas can easily escape bullet seed. Ivysaur has a horrible recovery and is easily edgeguarded.
Jiggly, advantage: Lucas. Jigglenaught doesn't have as much knockout potential. PK Thunder is hard to evade with a floaty character like Jiggles.
Kirby, advantage: ??? I haven't played this matchup enough to comment. There aren't enough Kirby players.
Link, advantage: Link. Link wins the projectile war, forcing Lucas to close range. Although the characters are roughly even in terms of power, Link will be able to manipulate the game more with his projectile spamming power.
Lucario, advantage: Lucas. Lucario's d-air is out prioritized by both PK Thunder and Lucas's u-smash. The dark orb can be absorbed, replenishing a huge amount of health. Lucario gets smashed up easily and Lucas smashes up very hard.
Mario, advantage: ??? Again, I haven't played this one enough.
Ness, advantage: Lucas. Lucas has better attacks. Lucas's psi magnet is a great attack. Ness's PK Fire can destroy Lucas, but most projectiles in this match will end up getting absorbed anyway. Lucas also has a ranged throw and powerful smashes (the yo-yo can't compare against the hexagons).
Peach, advantage: ??? Peach doesn't seem to get much play since she got balanced from Melee.
Pikachu, advantage: ??? I play against a lot of people who play bad Pikachus. Pikachu shouldn't be that bad against Lucas, although Lucas can PK Thunder 2 through any lightning attack.
Pikmin/Olimar, advantage: Olimar. Pikmin thrown cause PK Fire to fail. Olimar can easily grab, has long ranged smashes, and a good ranged game. Lucas will be forced to go on the offensive and won't perform well.
Pit, advantage: Pit. Pit has a ridiculously powerful projectile. His shield (and possibly his circular saw) deflect spells. Pit's smashes are fast, hard hitting, and have nearly no lag after them. Lucas will have problems.
R.O.B., advantage: neutral. ROB can punish Lucas with lasers and gyros, but Lucas can play the ranged game just as well, if not better. ROB has long ranged aerials that will destroy Lucas if he gets stuck in the air. ROB's d-smash murders Lucas. Lucas shouldn't have any trouble hitting ROB, since he's massive.
Samus, advantage: ??? You can absorb the charge shot, but most Samus players just spam the smash missiles. I don't know how this would play out.
Shiek, advantage: neutral. Shiek is fast and still powerful (although, no longer god-tier). Shiek's biggest problem will be getting around the fire and thunder spells. Lucas's stick smash and rope snake grab will be useful here.
Snake, advantage: Lucas. Snake might have issues closing distance. Moves like Psi Magnet will keep Snake away. PK Thunder can detonate claymores. Lucas can easily punish Snake's recovery. Although Snake's melee combo is hard hitting, it can also be prevented with fire spamming.
Sonic, advantage: ??? No idea, but this one seems neutral. Sonic's roll has lousy priority.
Squirtle, advantage: Squirtle. The Squirtle will have to play **** good to beat a Lucas, but he is fast enough to do so. Squirtle is fast enough to continuously deal aerial attacks to knock Lucas out and evade spells. Lucas has a huge delay after failing to land u-smash or a throw that Squirtle can jump all over and punish.
Toon Link, advantage: Toon Link. TL has sick projectiles. Lucas can't absorb anything, so he's once again forced to close range. People who can short hop TL's b-air will slaughter Lucas.
Wario, advantage: neutral. Wario has a lot of things going for him. Easy ways to get damage, stupid high priority with bite, insane air control, massive recovery, heavy weight, an instant kill skill every minute, and good smash attacks. He's also fat, an easy target, short ranged, and low priority on everything else he does. Warios will have to play a careful game, probably with lots of d-air d-smash hits.
Wolf, advantage: Wolf. Projectile deflection, a mean ranged attack, and even meaner melee attacks. Unless the person using Wolf doesn't know his character, Lucas is in for pain.
Yoshi, advantage: ??? Who plays this goofy dino?
Zelda, advantage: Lucas. Din's Fire is useless here, so Zelda will have to go with a different plan. A Zelda that knows her tilts should be able to play just fine against Lucas. Fortunately for Lucas players, 90% of Zelda users don't care about her tilts and can't land many air kicks. The Zelda player would have to have madd skills to land any of her awesome air kicks.
ZSS, advantage: ??? I play a lot of ZSS, but I haven't seen this matchup enough to comment.